Essentials of Analysis III PDF

MATH 85 Essentials of Analysis III Genrev Josiah A. Villamin, MSc [email protected] Department of Physical Sciences and Mathematics College of Arts and Science, University of the Philippines Manila Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals ...

MATH 85 Essentials of Analysis III Genrev Josiah A. Villamin, MSc [email protected] Department of Physical Sciences and Mathematics College of Arts and Science, University of the Philippines Manila Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals  Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.1. Directional Derivative of a Function). Let f be a function of two variables x and y. If U is the unit vector cos θi + sin θj, then the directional derivative of f in the direction of U, denoted by DU f, is given by f(x + h cos θ, y + h sin θ) − f(x, y) DU f(x, y) = lim , if this limit exists. h→0 h Prepared by GJ Villamin, 2024 fi  Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.1. Directional Derivative of a Function). Let f be a function of two variables x and y. If U is the unit vector cos θi + sin θj, then the directional derivative of f in the direction of U, denoted by DU f, is given by f(x + h cos θ, y + h sin θ) − f(x, y) DU f(x, y) = lim , if this limit exists. h→0 h The directional derivative gives the rate of change of the function values f(x, y) with respect to the direction of the unit vector U. Prepared by GJ Villamin, 2024 fi  Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.1. Directional Derivative of a Function). Let f be a function of two variables x and y. If U is the unit vector cos θi + sin θj, then the directional derivative of f in the direction of U, denoted by DU f, is given by f(x + h cos θ, y + h sin θ) − f(x, y) DU f(x, y) = lim , if this limit exists. h→0 h The directional derivative gives the rate of change of the function values f(x, y) with respect to the direction of the unit vector U. The partial derivatives fx and fy are then special cases of the directional derivative in the directions of the unit vectors i and j, respectively. Prepared by GJ Villamin, 2024 fi Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals ⟨ 6⟩ 2 2 π π Example: f(x, y) = 3x − y + 4x, U = cos , sin 6 Prepared by GJ Villamin, 2024 Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals ⟨ 6⟩ 2 2 π π Example: f(x, y) = 3x − y + 4x, U = cos , sin 6 ( 2) 3h h f x+ 2 ,y + − f(x, y) D U f(x, y) = lim h→0 h Prepared by GJ Villamin, 2024 Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals ⟨ 6⟩ 2 2 π π Example: f(x, y) = 3x − y + 4x, U = cos , sin 6 ( 2) 3h h f x+ 2 ,y + − f(x, y) D U f(x, y) = lim h→0 h ( ) ( ) 2 − (y + 2) 3h 2 3h h 2 2 3 x+ 2 +4 x+ 2 − 3x + y − 4x = lim h→0 h Prepared by GJ Villamin, 2024 Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals ⟨ 6⟩ 2 2 π π Example: f(x, y) = 3x − y + 4x, U = cos , sin 6 ( 2) 3h h f x+ 2 ,y + − f(x, y) DU f(x, y) = lim h→0 h ( ) ( ) 2 − (y + 2) 3h 2 3h h 2 2 3 x+ 2 +4 x+ 2 − 3x + y − 4x = lim h→0 h 2 9h 2 2 h2 2 2 3x + 3 3xh + 4 − y − yh − 4 + 4x + 2 3h − 3x + y − 4x = lim h→0 h Prepared by GJ Villamin, 2024 Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals ⟨ 6⟩ 2 2 π π Example: f(x, y) = 3x − y + 4x, U = cos , sin 6 ( 2) 3h h f x+ 2 ,y + − f(x, y) DU f(x, y) = h→0 lim h ( ) ( ) 2 − (y + 2) 3h 2 3h h 2 2 3 x+ 2 +4 x+ 2 − 3x + y − 4x = lim h→0 h 9h 2 h2 3x 2 + 3 3xh + 4 − y 2 − yh − 4 + 4x + 2 3h − 3x 2 + y 2 − 4x = lim h→0 h 2 3 3xh + 2h − yh + 2 3h = lim h→0 h Prepared by GJ Villamin, 2024 Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals ⟨ 6⟩ 2 2 π π Example: f(x, y) = 3x − y + 4x, U = cos , sin 6 ( 2) 3h h f x+ 2 ,y + − f(x, y) DU f(x, y) = h→0 lim h ( ) ( ) 2 − (y + 2) 3h 2 3h h 3 x+ 2 +4 x+ 2 − 3x 2 + y 2 − 4x = lim h→0 h 2 9h 2 2 h2 3x + 3 3xh + 4 − y − yh − 4 + 4x + 2 3h − 3x 2 + y 2 − 4x = lim h→0 h 3 3xh + 2h 2 − yh + 2 3h = lim h→0 h = lim (3 3x + 2h − y + 2 3 ) h→0 Prepared by GJ Villamin, 2024 Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: f(x, y) = 3x − y + 4x, U = ⟨cos 6 , sin 6 ⟩ 2 2 π π ( 2) 3h h f x+ 2 ,y + − f(x, y) DU f(x, y) = h→0 lim h ( ) ( ) 2 − (y + 2) 3h 2 3h h 3 x+ 2 +4 x+ 2 − 3x 2 + y 2 − 4x = lim h→0 h 2 9h 2 2 h2 3x + 3 3xh + 4 − y − yh − 4 + 4x + 2 3h − 3x 2 + y 2 − 4x = lim h→0 h 3 3xh + 2h 2 − yh + 2 3h = lim h→0 h = lim (3 3x + 2h − y + 2 3 ) h→0 = 3 3x − y + 2 3 Prepared by GJ Villamin, 2024 Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (Theorem 3.2.). If f is a di erentiable function of x and y, and U = cos θi + sin θj, then DU f(x, y) = fx(x, y)cos θ + fy(x, y)sin θ. Prepared by GJ Villamin, 2024 ff Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (Theorem 3.2.). If f is a di erentiable function of x and y, and U = cos θi + sin θj, then DU f(x, y) = fx(x, y)cos θ + fy(x, y)sin θ. ⟨ 6⟩ 2 2 π π Illustration: f(x, y) = 3x − y + 4x, U = cos , sin 6 Prepared by GJ Villamin, 2024 ff Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (Theorem 3.2.). If f is a di erentiable function of x and y, and U = cos θi + sin θj, then DU f(x, y) = fx(x, y)cos θ + fy(x, y)sin θ. ⟨ 6⟩ 2 2 π π Illustration: f(x, y) = 3x − y + 4x, U = cos , sin 6 ( 6) ( 6) π π D U f(x, y) = (6x + 4) cos − 2y sin Prepared by GJ Villamin, 2024 ff Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (Theorem 3.2.). If f is a di erentiable function of x and y, and U = cos θi + sin θj, then DU f(x, y) = fx(x, y)cos θ + fy(x, y)sin θ. ⟨ 6⟩ 2 2 π π Illustration: f(x, y) = 3x − y + 4x, U = cos , sin 6 ( 6) ( 6) π π DU f(x, y) = (6x + 4) cos − 2y sin ( 2 ) (2) 3 1 = (6x + 4) − 2y Prepared by GJ Villamin, 2024 ff Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (Theorem 3.2.). If f is a di erentiable function of x and y, and U = cos θi + sin θj, then DU f(x, y) = fx(x, y)cos θ + fy(x, y)sin θ. ⟨ 6⟩ 2 2 π π Illustration: f(x, y) = 3x − y + 4x, U = cos , sin 6 ( 6) ( 6) π π DU f(x, y) = (6x + 4) cos − 2y sin ( 2 ) (2) 3 1 = (6x + 4) − 2y = 3 3x − y + 2 3 Prepared by GJ Villamin, 2024 ff Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (Theorem 3.2.). If f is a di erentiable function of x and y, and U = cos θi + sin θj, then DU f(x, y) = fx(x, y)cos θ + fy(x, y)sin θ. The directional derivative can be written as the dot product of two vectors DU f(x, y) = ( fx(x, y)i + fy(x, y)j) ⋅ (cos θi + sin θj) = ∇f(x, y) ⋅ U. Prepared by GJ Villamin, 2024 ff  Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.3. Gradient of a Function). If f is a function of two variables x and y and fx and fy exist, then the gradient of f, denoted by ∇f, is de ned by ∇f(x, y) = fx(x, y)i + fy(x, y)j. Prepared by GJ Villamin, 2024 fi fi  Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.3. Gradient of a Function). If f is a function of two variables x and y and fx and fy exist, then the gradient of f, denoted by ∇f, is de ned by ∇f(x, y) = fx(x, y)i + fy(x, y)j. If α is the radian measure of the angle between the two vectors ∇f and U, then DU f = ∇f ⋅ U = | ∇f | | U | cos α = | ∇f | cos α. Prepared by GJ Villamin, 2024 fi fi  Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.3. Gradient of a Function). If f is a function of two variables x and y and fx and fy exist, then the gradient of f, denoted by ∇f, is de ned by ∇f(x, y) = fx(x, y)i + fy(x, y)j. If α is the radian measure of the angle between the two vectors ∇f and U, then DU f = ∇f ⋅ U = | ∇f | | U | cos α = | ∇f | cos α. DU f = | ∇f | will be a maximum when cos α = 1, that is, when U is in the direction of ∇f. Prepared by GJ Villamin, 2024 fi fi  Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.3. Gradient of a Function). If f is a function of two variables x and y and fx and fy exist, then the gradient of f, denoted by ∇f, is de ned by ∇f(x, y) = fx(x, y)i + fy(x, y)j. If α is the radian measure of the angle between the two vectors ∇f and U, then DU f = ∇f ⋅ U = | ∇f | | U | cos α = | ∇f | cos α. DU f = | ∇f | will be a maximum when cos α = 1, that is, when U is in the direction of ∇f. Similarly, DU f = − | ∇f | will be a minimum when cos α = − 1, that is, when U is in the opposite direction of ∇f. Prepared by GJ Villamin, 2024 fi fi  Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 2 2 x y Example: If f(x, y) = + , nd the gradient of f at the point (4,3). Also 16 9 π nd the rate of change of f(x, y) in the direction at (4,3). 4 Prepared by GJ Villamin, 2024 fi fi  Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 2 2 x y Example: If f(x, y) = + , nd the gradient of f at the point (4,3). Also 16 9 π nd the rate of change of f(x, y) in the direction at (4,3). 4 1 2 ∇f(x, y) = xi + yj 8 9 Prepared by GJ Villamin, 2024 fi fi  Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 2 2 x y Example: If f(x, y) = + , nd the gradient of f at the point (4,3). Also 16 9 π nd the rate of change of f(x, y) in the direction at (4,3). 4 1 2 ∇f(x, y) = xi + yj 8 9 1 2 ∇f(4,3) = xi + yj 2 3 Prepared by GJ Villamin, 2024 fi fi Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 2 2 x y Example: If f(x, y) = + , nd the gradient of f at the point (4,3). Also nd the rate of 16 9 π change of f(x, y) in the direction at (4,3). 4 1 2 ∇f(x, y) = xi + yj 8 9 1 2 ∇f(4,3) = xi + yj 2 3 ⟨2 3⟩ ⟨ 2 2 ⟩ 1 2 2 2 DU f(4,3) = , ⋅ , Prepared by GJ Villamin, 2024 fi fi Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 2 2 x y Example: If f(x, y) = + , nd the gradient of f at the point (4,3). Also nd the rate of change of f(x, y) in the π 16 9 direction at (4,3). 4 1 2 ∇f(x, y) = xi + yj 8 9 1 2 ∇f(4,3) = 2 xi + 3 yj ⟨2 3⟩ ⟨ 2 2 ⟩ 1 2 2 2 DU f(4,3) = , ⋅ , 7 2 DU f(4,3) = 12 Prepared by GJ Villamin, 2024 fi fi Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 2 2 Example: Given f(x, y) = 2x − y + 3x − y, nd the maximum value of DU f at the point where x = 1 and y = − 2. Prepared by GJ Villamin, 2024 fi Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 2 2 Example: Given f(x, y) = 2x − y + 3x − y, nd the maximum value of DU f at the point where x = 1 and y = − 2. ∇f(x, y) = (4x + 3)i + (−2y − 1)j Prepared by GJ Villamin, 2024 fi Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 2 2 Example: Given f(x, y) = 2x − y + 3x − y, nd the maximum value of DU f at the point where x = 1 and y = − 2. ∇f(x, y) = (4x + 3)i + (−2y − 1)j ∇f(1, − 2) = 7i + 3j Prepared by GJ Villamin, 2024 fi Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 2 2 Example: Given f(x, y) = 2x − y + 3x − y, nd the maximum value of DU f at the point where x = 1 and y = − 2. ∇f(x, y) = (4x + 3)i + (−2y − 1)j ∇f(1, − 2) = 7i + 3j | ∇f(1, − 2) | = 49 + 9 = 58 Prepared by GJ Villamin, 2024 fi Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: The temperature at any point (x, y) of a rectangular plate lying in the 2 2 xy plane is determined by T(x, y) = x + y. Find the rate of change of the temperature at the point (3,4) in the direction making an angle of radian π measure with the positive x direction. 3 Prepared by GJ Villamin, 2024 Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: The temperature at any point (x, y) of a rectangular plate lying in the 2 2 xy plane is determined by T(x, y) = x + y. Find the rate of change of the temperature at the point (3,4) in the direction making an angle of radian π measure with the positive x direction. 3 ( 3) ( 3) π π D U T(3,4) = Tx (3,4) cos + T y (3,4) sin Prepared by GJ Villamin, 2024 Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: The temperature at any point (x, y) of a rectangular plate lying in the 2 2 xy plane is determined by T(x, y) = x + y. Find the rate of change of the temperature at the point (3,4) in the direction making an angle of radian π measure with the positive x direction. 3 ( 3) ( 3) π π D U T(3,4) = Tx (3,4) cos + T y (3,4) sin = 3 + 4 3 Prepared by GJ Villamin, 2024 Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: The temperature at any point (x, y) of a rectangular plate lying in the xy plane is 2 2 determined by T(x, y) = x + y. Find the direction for which the rate of change of the temperature at the point (−3,1) is a maximum. ∇T(x, y) = 2xi + 2yj ∇T(−3,1) = − 6i + 2j 1 tan α = − 3 −1 1 α = π − tan 3 Prepared by GJ Villamin, 2024  Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals In three dimensional space, the direction of a vector is determined by its direction cosines. Let cos α, cos β, and cos γ be the direction cosines of the unit vector U, so that U = cos αi + cos βj + cos γk. (De nition 3.4. Directional Derivative of a Function). Suppose that f is a function of three variables x, y, and z. If U is the unit vector U = cos αi + cos βj + cos γk, then the directional derivative of f in the direction of U, denoted by DU f, is given f(x + h cos α, y + h cos β, z + h cos γ) − f(x, y, z) by DU f(x, y, z) = lim , if this h→0 h limit exists. Prepared by GJ Villamin, 2024 fi Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals In three dimensional space, the direction of a vector is determined by its direction cosines. Let cos α, cos β, and cos γ be the direction cosines of the unit vector U, so that U = cos αi + cos βj + cos γk. (Theorem 3.5.). If f is a di erentiable function of x, y, and z, and U = cos αi + cos βj + cos γk, then DU f(x, y, z) = fx(x, y, z)cos α + fy(x, y, z)cos β + fz(x, y, z)cos γ. Prepared by GJ Villamin, 2024 ff Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 2 2 2 Illustration: Given f(x, y, z) = 3x + xy − 2y − yz + z , nd the rate of change of f(x, y, z) at (1, − 2, − 1) in the direction of the vector 2i − 2j − k. (3) ( 3) ( 3) 2 2 1 D U f(x, y, z) = (6x + y) + (x − 4y − z) − + (−y + 2z) − (3) ( 3) ( 3) 2 2 1 D U f(1, − 2, − 1) = 4 + 10 − + 0 − = − 4 Prepared by GJ Villamin, 2024 fi  Directional Derivative and Gradient Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.6. Gradient of a Function). If f is a function of three variables x, y, and z, and fx, fy, and fz exist, then the gradient of f, denoted by ∇f, is de ned by ∇f(x, y, z) = fx(x, y, z)i + fy(x, y, z)j + fz(x, y, z)k. Prepared by GJ Villamin, 2024 fi fi  Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.7. Normal Vector). A vector which is orthogonal to the unit tangent vector at every curve C through a point P0 on a surface S is called a normal vector to S at P0. Prepared by GJ Villamin, 2024 fi  Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.7. Normal Vector). A vector which is orthogonal to the unit tangent vector at every curve C through a point P0 on a surface S is called a normal vector to S at P0. (Theorem 3.8.). If an equation of a surface S is F(x, y, z) = 0, and Fx, Fy, and Fz are continuous and not all zero at the point P0(x0, y0, z0) on S, then a normal vector to S at P0 is ∇F(x0, y0, z0). Prepared by GJ Villamin, 2024 fi  Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.7. Normal Vector). A vector which is orthogonal to the unit tangent vector at every curve C through a point P0 on a surface S is called a normal vector to S at P0. (Theorem 3.8.). If an equation of a surface S is F(x, y, z) = 0, and Fx, Fy, and Fz are continuous and not all zero at the point P0(x0, y0, z0) on S, then a normal vector to S at P0 is ∇F(x0, y0, z0). (De nition 3.9. Tangent Plane). If an equation of a surface is F(x, y, z) = 0, then the tangent plane of S at a point P0(x0, y0, z0) is the plane through P0 having as a normal vector ∇F(x0, y0, z0). Prepared by GJ Villamin, 2024 fi fi  Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.7. Normal Vector). A vector which is orthogonal to the unit tangent vector at every curve C through a point P0 on a surface S is called a normal vector to S at P0. (Theorem 3.8.). If an equation of a surface S is F(x, y, z) = 0, and Fx, Fy, and Fz are continuous and not all zero at the point P0(x0, y0, z0) on S, then a normal vector to S at P0 is ∇F(x0, y0, z0). (De nition 3.9. Tangent Plane). If an equation of a surface is F(x, y, z) = 0, then the tangent plane of S at a point P0(x0, y0, z0) is the plane through P0 having as a normal vector ∇F(x0, y0, z0). An equation of the tangent plane of the above de nition is Fx(x0, y0, z0)(x − x0) + Fy(x0, y0, z0)(y − y0) + Fz(x0, y0, z0)(z − z0) = 0. Prepared by GJ Villamin, 2024 fi fi fi  Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.7. Normal Vector). A vector which is orthogonal to the unit tangent vector at every curve C through a point P0 on a surface S is called a normal vector to S at P0. (Theorem 3.8.). If an equation of a surface S is F(x, y, z) = 0, and Fx, Fy, and Fz are continuous and not all zero at the point P0(x0, y0, z0) on S, then a normal vector to S at P0 is ∇F(x0, y0, z0). (De nition 3.9. Tangent Plane). If an equation of a surface is F(x, y, z) = 0, then the tangent plane of S at a point P0(x0, y0, z0) is the plane through P0 having as a normal vector ∇F(x0, y0, z0). An equation of the tangent plane of the above de nition is Fx(x0, y0, z0)(x − x0) + Fy(x0, y0, z0)(y − y0) + Fz(x0, y0, z0)(z − z0) = 0. A vector equation of the tangent plane is ∇F(x0, y0, z0) ⋅ ((x − x0)i + (y − y0)j + (z − z0)k) = 0. Prepared by GJ Villamin, 2024 fi fi fi Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find an equation of the tangent plane to the elliptic paraboloid 2 2 4x + y − 16z = 0 at the point (2,4,2). Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find an equation of the tangent plane to the elliptic paraboloid 2 2 4x + y − 16z = 0 at the point (2,4,2). ∇F(x, y, z) = 8xi + 2yj − 16k Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find an equation of the tangent plane to the elliptic paraboloid 2 2 4x + y − 16z = 0 at the point (2,4,2). ∇F(x, y, z) = 8xi + 2yj − 16k ∇F(2,4,2) = 16i + 8j − 16k Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find an equation of the tangent plane to the elliptic paraboloid 2 2 4x + y − 16z = 0 at the point (2,4,2). ∇F(x, y, z) = 8xi + 2yj − 16k ∇F(2,4,2) = 16i + 8j − 16k 16(x − 2) + 8(y − 4) − 16(z − 2) = 0 Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find an equation of the tangent plane to the elliptic paraboloid 2 2 4x + y − 16z = 0 at the point (2,4,2). ∇F(x, y, z) = 8xi + 2yj − 16k ∇F(2,4,2) = 16i + 8j − 16k 16(x − 2) + 8(y − 4) − 16(z − 2) = 0 2x + y − 2z − 4 = 0 Prepared by GJ Villamin, 2024  Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.10. Normal Line). The normal line to a surface S at a point P0 on S is the line through P0 having as a set of direction numbers the components of any normal vector to S at P0. Prepared by GJ Villamin, 2024 fi  Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.10. Normal Line). The normal line to a surface S at a point P0 on S is the line through P0 having as a set of direction numbers the components of any normal vector to S at P0. If an equation of a surface S is F(x, y, z)x = 0, then symmetric equations of the − x0 y − y0 z − z0 normal line to S at P0(x0, y0, z0) are = =. Fx(x0, y0, z0) Fy(x0, y0, z0) Fz(x0, y0, z0) Prepared by GJ Villamin, 2024 fi Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find symmetric equations of the normal line to the surface 2 2 4x + y − 16z = 0 at the point (2,4,2). Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find symmetric equations of the normal line to the surface 2 2 4x + y − 16z = 0 at the point (2,4,2). x−2 y−4 z−2 2 = = 1 −2 Prepared by GJ Villamin, 2024  Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.11. Tangent Line). The tangent line to a curve C at a point P0 is the line through P0 having as a set of direction numbers the components of the unit tangent vector to C at P0. Prepared by GJ Villamin, 2024 fi Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find symmetric equations of the tangent line to the curve of 2 2 2 2 2 2 intersection of the surfaces 3x + 2y + z = 49 and x + y − 2z = 10 at the point (3, − 3,2). Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find symmetric equations of the tangent line to the curve of 2 2 2 2 2 2 intersection of the surfaces 3x + 2y + z = 49 and x + y − 2z = 10 at the point (3, − 3,2). 2 2 2 Let F(x, y, z) = 3x + 2y + z − 49. Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find symmetric equations of the tangent line to the curve of 2 2 2 2 2 2 intersection of the surfaces 3x + 2y + z = 49 and x + y − 2z = 10 at the point (3, − 3,2). 2 2 2 Let F(x, y, z) = 3x + 2y + z − 49. ∇F(x, y, z) = 6xi + 4yj + 2zk Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find symmetric equations of the tangent line to the curve of 2 2 2 2 2 2 intersection of the surfaces 3x + 2y + z = 49 and x + y − 2z = 10 at the point (3, − 3,2). 2 2 2 Let F(x, y, z) = 3x + 2y + z − 49. ∇F(x, y, z) = 6xi + 4yj + 2zk ∇F(3, − 3,2) = 18i − 12j + 4k = 2(9i − 6j + 2k) Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find symmetric equations of the tangent line to the curve of 2 2 2 2 2 2 intersection of the surfaces 3x + 2y + z = 49 and x + y − 2z = 10 at the point (3, − 3,2). 2 2 2 Let G(x, y, z) = x + y − 2z − 10. Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find symmetric equations of the tangent line to the curve of 2 2 2 2 2 2 intersection of the surfaces 3x + 2y + z = 49 and x + y − 2z = 10 at the point (3, − 3,2). 2 2 2 Let G(x, y, z) = x + y − 2z − 10. ∇G(x, y, z) = 2xi + 2yj − 4zk Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find symmetric equations of the tangent line to the curve of 2 2 2 2 2 2 intersection of the surfaces 3x + 2y + z = 49 and x + y − 2z = 10 at the point (3, − 3,2). 2 2 2 Let G(x, y, z) = x + y − 2z − 10. ∇G(x, y, z) = 2xi + 2yj − 4zk ∇G(3, − 3,2) = 6i − 6j − 8k = 2(3i − 3j − 4k) Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find symmetric equations of the tangent line to the curve of 2 2 2 2 2 2 intersection of the surfaces 3x + 2y + z = 49 and x + y − 2z = 10 at the point (3, − 3,2). ∇F(3, − 3,2) × ∇G(3, − 3,2) = ⟨9, − 6,2⟩ × ⟨3, − 3, − 4⟩ Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find symmetric equations of the tangent line to the curve of 2 2 2 2 2 2 intersection of the surfaces 3x + 2y + z = 49 and x + y − 2z = 10 at the point (3, − 3,2). ∇F(3, − 3,2) × ∇G(3, − 3,2) = ⟨9, − 6,2⟩ × ⟨3, − 3, − 4⟩ = < 10,14, − 3 > Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find symmetric equations of the tangent line to the curve of 2 2 2 2 2 2 intersection of the surfaces 3x + 2y + z = 49 and x + y − 2z = 10 at the point (3, − 3,2). ∇F(3, − 3,2) × ∇G(3, − 3,2) = ⟨9, − 6,2⟩ × ⟨3, − 3, − 4⟩ = < 10,14, − 3 > x−3 y+3 z−2 10 = = 14 −3 Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find parametric equations of the tangent line to the curve of intersection of the surfaces x = 2 + cos πyz and y = 1 + sin πxz at the point (3,1,2). Let F(x, y, z) = x − 2 − cos πyz. Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find parametric equations of the tangent line to the curve of intersection of the surfaces x = 2 + cos πyz and y = 1 + sin πxz at the point (3,1,2). Let F(x, y, z) = x − 2 − cos πyz. ∇F(x, y, z) = i + πz sin πyzj + πy sin πyzk Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find parametric equations of the tangent line to the curve of intersection of the surfaces x = 2 + cos πyz and y = 1 + sin πxz at the point (3,1,2). Let F(x, y, z) = x − 2 − cos πyz. ∇F(x, y, z) = i + πz sin πyzj + πy sin πyzk ∇F(3,1,2) = i + 2π sin 2πj + π sin 2πk = i Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find parametric equations of the tangent line to the curve of intersection of the surfaces x = 2 + cos πyz and y = 1 + sin πxz at the point (3,1,2). Let F(x, y, z) = x − 2 − cos πyz. Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find parametric equations of the tangent line to the curve of intersection of the surfaces x = 2 + cos πyz and y = 1 + sin πxz at the point (3,1,2). Let F(x, y, z) = x − 2 − cos πyz. ∇G(x, y, z) = − πz cos πxzi + j − πx cos πxzk Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find parametric equations of the tangent line to the curve of intersection of the surfaces x = 2 + cos πyz and y = 1 + sin πxz at the point (3,1,2). Let F(x, y, z) = x − 2 − cos πyz. ∇G(x, y, z) = − πz cos πxzi + j − πx cos πxzk ∇G(3,1,2) = − 2π cos 6πi + j − 3π cos 6πk = − 2πi + j − 3πk Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find parametric equations of the tangent line to the curve of intersection of the surfaces x = 2 + cos πyz and y = 1 + sin πxz at the point (3,1,2). ∇F(3,1,2) × ∇G(3,1,2) = ⟨1,0,0⟩ × ⟨−2π,1, − 3π⟩ = ⟨0,3π,1⟩ Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find parametric equations of the tangent line to the curve of intersection of the surfaces x = 2 + cos πyz and y = 1 + sin πxz at the point (3,1,2). ∇F(3,1,2) × ∇G(3,1,2) = ⟨1,0,0⟩ × ⟨−2π,1, − 3π⟩ = ⟨0,3π,1⟩ x(t) = 3 Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find parametric equations of the tangent line to the curve of intersection of the surfaces x = 2 + cos πyz and y = 1 + sin πxz at the point (3,1,2). ∇F(3,1,2) × ∇G(3,1,2) = ⟨1,0,0⟩ × ⟨−2π,1, − 3π⟩ = ⟨0,3π,1⟩ x(t) = 3 y(t) = 1 + (3π)t Prepared by GJ Villamin, 2024 Tangent Planes and Normals to Surfaces Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Example: Find parametric equations of the tangent line to the curve of intersection of the surfaces x = 2 + cos πyz and y = 1 + sin πxz at the point (3,1,2). ∇F(3,1,2) × ∇G(3,1,2) = ⟨1,0,0⟩ × ⟨−2π,1, − 3π⟩ = ⟨0,3π,1⟩ x(t) = 3 y(t) = 1 + (3π)t z(t) = 2 + t Prepared by GJ Villamin, 2024  Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.12. Absolute Maximum Value). The function f of two variables is said to have an absolute maximum value on a disk B in the xy plane if there is some point (x0, y0) in B such that f(x0, y0) ≥ f(x, y) for all points (x, y) in B. In such case, f(x0, y0) is the absolute maximum value. Prepared by GJ Villamin, 2024 fi  Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.12. Absolute Maximum Value). The function f of two variables is said to have an absolute maximum value on a disk B in the xy plane if there is some point (x0, y0) in B such that f(x0, y0) ≥ f(x, y) for all points (x, y) in B. In such case, f(x0, y0) is the absolute maximum value. (De nition 3.13. Absolute Minimum Value). The function f of two variables is said to have an absolute minimum value on a disk B in the xy plane if there is some point (x0, y0) in B such that f(x0, y0) ≤ f(x, y) for all points (x, y) in B. In such case, f(x0, y0) is the absolute minimum value. Prepared by GJ Villamin, 2024 fi fi Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (Theorem 3.14. Extreme Value Theorem). Let B be a closed disk in the xy plane, and let f be a function of two variables which is continuous on B. Then there is at least one point in B where f has an absolute maximum value and at least one point in B where f has an absolute minimum value. Prepared by GJ Villamin, 2024  Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.15. Relative Maximum Value). The function f of two variables is said to have a relative maximum value at the point (x0, y0) if there exists an open disk B((x0, y0), r) such that f(x, y) ≤ f(x0, y0) for all (x, y) in the open disk. Prepared by GJ Villamin, 2024 fi  Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.15. Relative Maximum Value). The function f of two variables is said to have a relative maximum value at the point (x0, y0) if there exists an open disk B((x0, y0), r) such that f(x, y) ≤ f(x0, y0) for all (x, y) in the open disk. (De nition 3.16. Relative Minimum Value). The function f of two variables is said to have a relative minimum value at the point (x0, y0) if there exists an open disk B((x0, y0), r) such that f(x, y) ≥ f(x0, y0) for all (x, y) in the open disk. Prepared by GJ Villamin, 2024 fi fi Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (Theorem 3.17.). If f(x, y) exists at all points in some open disk B((x0, y0), r) and if f has a relative extremum at (x0, y0), then if fx(x0, y0) and fy(x0, y0) exist, fx(x0, y0) = fy(x0, y0) = 0. Prepared by GJ Villamin, 2024  Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.18. Critical Point). A point (x0, y0) for which both fx(x0, y0) = 0 and fy(x0, y0) = 0 is called a critical point. Prepared by GJ Villamin, 2024 fi  Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.18. Critical Point). A point (x0, y0) for which both fx(x0, y0) = 0 and fy(x0, y0) = 0 is called a critical point. It is possible for a function of two variables to have a relative extremum at a point at which the partial derivatives do not exist. Prepared by GJ Villamin, 2024 fi  Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.18. Critical Point). A point (x0, y0) for which both fx(x0, y0) = 0 and fy(x0, y0) = 0 is called a critical point. It is possible for a function of two variables to have a relative extremum at a point at which the partial derivatives do not exist. The vanishing of the rst partial derivatives of a function of two variables is not a su cient condition for the function to have a relative extremum at the point. Such a situation occurs at a point called a saddle point. 2 2 Example: f(x, y) = y − x Prepared by GJ Villamin, 2024 fi ffi fi  Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (Theorem 3.19. Second Derivative Test). Let f be a function of two variables such that f and its rst and second order partial derivatives are continuous on some open disk B((x0, y0), r). Suppose further that fx(x0, y0) = fy(x0, y0) = 0. Then: 2 f has a relative minimum value at (x0, y0) if fxx(x0, y0)fyy(x0, y0) − fxy (x0, y0) > 0 and fxx(x0, y0) > 0 2 f has a relative maximum value at (x0, y0) if fxx(x0, y0)fyy(x0, y0) − fxy (x0, y0) > 0 and fxx(x0, y0) < 0 2 f(x0, y0) is not a relative extremum if fxx(x0, y0)fyy(x0, y0) − fxy (x0, y0) 0 Prepared by GJ Villamin, 2024 Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 2 2 Illustration: f(x, y) = 6x − 4y − x − 2y fxx(3, − 1) = − 2 fyy(3, − 1) = − 4 2 fxy (3, − 1) = 0 2 fxx(3, − 1)fyy(3, − 1) − fxy (3, − 1) = (−2)(−4) − 0 = 8 > 0 f(3, − 1) is a relative maximum. Prepared by GJ Villamin, 2024 Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 4 2 2 Illustration: f(x, y) = 2x + y − x − 2y 3 1 1 x f (x, y) = 8x − 2x = 0 ⇒ x = − , x = 0, x = 2 2 fy(x, y) = 2y − 2 = 0 ⇒ y = 1 Prepared by GJ Villamin, 2024 Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 4 2 2 Illustration: f(x, y) = 2x + y − x − 2y 3 1 1 x f (x, y) = 8x − 2x = 0 ⇒ x = − , x = 0, x = 2 2 fy(x, y) = 2y − 2 = 0 ⇒ y = 1 ( 2 ) (2 ) 1 1 Critical points: − ,1 , (0,1), ,1 Prepared by GJ Villamin, 2024 Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 4 2 2 Illustration: f(x, y) = 2x + y − x − 2y ( 2 ) 1 fxx − ,1 = 4 ( 2 ) 1 fyy − ,1 = 2 ( 2 ) 2 1 fxy − ,1 = 0 Prepared by GJ Villamin, 2024 Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 4 2 2 Illustration: f(x, y) = 2x + y − x − 2y ( 2 ) 1 fxx − ,1 = 4 ( 2 ) 1 fyy − ,1 = 2 ( ) 2 1 fxy − ,1 =0 2 ( 2 ) ( 2 ) ( 2 ) 1 1 2 1 fxx − ,1 fyy − ,1 − fxy − ,1 = (4)(2) − 0 = 8 > 0 Prepared by GJ Villamin, 2024 Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Illustration: f(x, y) = 2x 4 + y 2 − x 2 − 2y ( 2 ) 1 fxx − ,1 = 4 ( 2 ) 1 fyy − ,1 = 2 ( 2 ) 2 1 f xy − ,1 = 0 ( 2 ) ( 2 ) ( 2 ) 1 1 2 1 fxx − ,1 fyy − ,1 − fxy − ,1 = (4)(2) − 0 = 8 > 0 ( 2 ) 1 f − ,1 is a relative minimum. Prepared by GJ Villamin, 2024 Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 4 2 2 Illustration: f(x, y) = 2x + y − x − 2y fxx(0,1) = − 2 fyy(0,1) = 2 2 fxy (0,1) =0 Prepared by GJ Villamin, 2024 Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 4 2 2 Illustration: f(x, y) = 2x + y − x − 2y fxx(0,1) = − 2 fyy(0,1) = 2 2 fxy (0,1) =0 2 fxx(0,1)fyy(0,1) − fxy (0,1) = (−2)(2) − 0 = − 4 < 0 Prepared by GJ Villamin, 2024 Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 4 2 2 Illustration: f(x, y) = 2x + y − x − 2y fxx(0,1) = − 2 fyy(0,1) = 2 2 fxy (0,1) = 0 2 fxx(0,1)fyy(0,1) − fxy (0,1) = (−2)(2) − 0 = − 4 < 0 f(0,1) is not a relative extremum. f(0,1) is a saddle point. Prepared by GJ Villamin, 2024 Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 4 2 2 Illustration: f(x, y) = 2x + y − x − 2y (2 ) 1 fxx ,1 = 4 (2 ) 1 fyy ,1 = 2 (2 ) 2 1 fxy ,1 = 0 Prepared by GJ Villamin, 2024 Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals 4 2 2 Illustration: f(x, y) = 2x + y − x − 2y (2 ) 1 fxx ,1 = 4 (2 ) 1 fyy ,1 = 2 ( ) 2 1 fxy ,1 =0 2 (2 ) (2 ) (2 ) 1 1 2 1 fxx ,1 fyy ,1 − fxy ,1 = (4)(2) − 0 = 8 > 0 Prepared by GJ Villamin, 2024 Extrema of Functions of Two Variables Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals Illustration: f(x, y) = 2x 4 + y 2 − x 2 − 2y (2 ) 1 fxx ,1 = 4 (2 ) 1 fyy ,1 = 2 (2 ) 2 1 f xy ,1 = 0 (2 ) (2 ) (2 ) 1 1 2 1 fxx ,1 fyy ,1 − fxy ,1 = (4)(2) − 0 = 8 > 0 (2 ) 1 f ,1 is a relative minimum. Prepared by GJ Villamin, 2024  Line Integrals Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (Theorem 3.20.). Suppose M and N are functions of two variables x and y 2 de ned on an open disk B((x0, y0), r) in ℝ , and My and Nx are continuous on B. Then the vector M(x, y)i + N(x, y)j is a gradient on B if and only if My(x, y) = Nx(x, y) at all points in B. (Theorem 3.21.). Suppose M, N, and R are functions of three variables x, y, and 3 z de ned on an open ball B((x0, y0, z0), r) in ℝ , and My, Mz, Nx, Nz, Rx, and Ry are continuous on B. Then the vector M(x, y)i + N(x, y)j + R(x, y)k is a gradient on B if and only if My(x, y, z) = Nx(x, y, z), Mz(x, y, z) = Rx(x, y, z), and Nz(x, y, z) = Ry(x, y, z) at all points in B. Prepared by GJ Villamin, 2024 fi fi  Line Integrals Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.22. Work). Let C be a curve lying in an open disk B in ℝ2 for which a vector equation of C is R(t) = f(t)i + g(t)j, where f′ and g′ are continuous on [a, b]. Furthermore, let a force eld be de ned on F(x, y) = M(x, y)i + N(x, y)j, where M and N are continuous on B. Then if W is the measure of work done by F in moving an object along C from ( f(a), g(a)) to ( f(b), g(b)), then: b ∫a W= (M( f(t), g(t))f′(t) + N( f(t), g(t))g′(t))dt b ∫a W= ⟨M( f(t), g(t)), N( f(t), g(t))⟩ ⋅ ⟨f′(t), g′(t)⟩dt b ∫a W= F( f(t), g(t)) ⋅ R′(t)dt ∫C This integral is called a line integral. A common notation for the line integral is M(x, y)dx + N(x, y)dy.        Prepared by GJ Villamin, 2024 fi fi fi  Line Integrals Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.23. Line Integral). Let M and N be functions of two variables x and y such that they are continuous on an open disk B in ℝ2. Let C be a curve lying in B and having parametric equations x = f(t), y = g(t), and a ≤ t ≤ b, such that f′ and g′ are continuous on [a, b]. Then the line integral of M(x, y)dx + N(x, y)dy over C, ∫C M(x, y)dx + N(x, y)dy, is given by: b ∫ ∫a (M( f(t), g(t))f′(t) + N( f(t), g(t))g′(t))dt = ⟨M( f(t), g(t)), N( f(t), g(t))⟩ ⋅ ⟨f′(t), g′(t)⟩dt a If an equation of C is of the form y = F(x), then x may be used as a parameter in place of t. Similarly, if an equation of C is of the form x = G(y), then y may be used as a parameter in place of t. If the curve C is the closed interval [a, b] on the x axis, then y = 0 and dy = 0. Thus, b ∫C ∫a M(x, y)dx + N(x, y)dy = M(x,0)dx.       Prepared by GJ Villamin, 2024 fi  Line Integrals Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.24. Smooth Curve). A curve C is said to be smooth if the derivatives f′ and g′ of the functions f and g in the parametric equations de ning C are continuous. If the curve C consists of a nite number of arcs of smooth curves, then C is said to be sectionally smooth. (De nition 3.25. Line Integral). Let the curve C consist of smooth arcs C1, C2, …, Cn. Then the line integral of M(x, y)dx + N(x, y)dy over C is de ned by: n ∫C ∑ (∫ ) M(x, y)dx + N(x, y)dy = M(x, y)dx + N(x, y)dy i=1 Ci   Prepared by GJ Villamin, 2024 fi fi fi fi fi  Line Integrals Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (De nition 3.26. Line Integral). Let M, N, and R be functions of three variables x, y, and z 3 such that they are continuous on an open ball B in R. Let C be a curve lying in B and having parametric equations x = f(t), y = g(t), z = h(t), and a ≤ t ≤ b, such that f′, g′, and h′ are continuous on [a, b]. Then the line integral of M(x, y, z)dx + N(x, y, z)dy + R(x, y, z)dz over ∫C C, M(x, y, z)dx + N(x, y, z)dy + R(x, y, z)dz, is given by: b ∫ (M( f(t), g(t), h(t))f′(t) + N( f(t), g(t), h(t))g′(t) + R( f(t), g(t), h(t))h′(t))dt a b ∫ ⟨M( f(t), g(t), h(t)), N( f(t), g(t), h(t)), R( f(t), g(t), h(t))⟩ ⋅ ⟨f′(t), g′(t), h′(t)⟩dt a          Prepared by GJ Villamin, 2024 fi Line Integrals Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (Theorem 3.27. Fundamental Theorem for Line Integrals). Suppose that M and N are functions of two variables x and y de ned on an open disk B((x0, y0), r) in 2 ℝ , My and Nx are continuous on B, and ∇ϕ(x, y) = M(x, y)i + N(x, y)j. Suppose that C is any sectionally smooth curve in B from the point (x1, y1) to the ∫C point (x2, y2). Then the line integral M(x, y)dx + N(x, y)dy is independent of ∫C the path C and M(x, y)dx + N(x, y)dy = ϕ(x2, y2) − ϕ(x1, y1). Prepared by GJ Villamin, 2024 fi Line Integrals Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals If F is the vector-valued function de ned by F(x, y) = M(x, y)i + N(x, y)j and F(x, y) = ∇ϕ(x, y), then F is called a gradient eld, ϕ is called a potential function, and ϕ(x, y) is the potential of F at (x, y). If F is a force eld satisfying F(x, y) = ∇ϕ(x, y), then F is said to be conservative. If F(x, y) = M(x, y)i + N(x, y)j has a potential function ϕ, then the expression M(x, y)dx + N(x, y)dy is called an exact di erential because M(x, y)dx + N(x, y)dy = dϕ. Prepared by GJ Villamin, 2024 fi fi fi ff Line Integrals Directional Derivatives, Gradients, Applications of Partial Derivatives, and Line Integrals (Theorem 3.28. Fundamental Theorem for Line Integrals). Suppose that M, N, and R are functions 3 of three variables x, y, and z de ned on an open ball B((x0, y0, z0), r) in ℝ , My, Mz, Nx, Nz, Ry, and Rz are continuous on B, and ∇ϕ(x, y, z) = M(x, y, z)i + N(x, y, z)j + R(x, y, z)k. Suppose that C is any sectionally smooth curve in B from the point (x1, y1, z1) to the point (x2, y2, z2). Then the line ∫C integral M(x, y, z)dx + N(x, y, z)dy + R(x, y, z)dz is independent of the path C and ∫C M(x, y, z)dx + N(x, y, z)dy + R(x, y, z)dz = ϕ(x2, y2, z2) − ϕ(x1, y1, z1). If F(x, y, z) = ∇ϕ(x, y, z), then F is a gradient eld, ϕ is a potential function, and ϕ(x, y, z) is the potential of F at (x, y, z). If F is a force eld having a potential function, then F is conservative. If F(x, y, z) = M(x, y, z)i + N(x, y, z)j + R(x, y, z)k has a potential function, then the expression M(x, y, z)dx + N(x, y, z)dy + R(x, y, z)dz is an exact di erential. Prepared by GJ Villamin, 2024 fi fi fi ff  References Di erential Calculus of Functions of Several Variables Leithold, L. The Calculus 7. Stewart, J. Calculus Early Transcendentals. Prepared by GJ Villamin, 2024 ff MATH 85 Essentials of Analysis III Genrev Josiah A. Villamin, MSc [email protected] Department of Physical Sciences and Mathematics College of Arts and Science, University of the Philippines Manila Differential Calculus of Functions of Several Variables  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.1. Distance). If P(x1, x2, …, xn) and Q(y1, y2, …, yn) are two n points in ℝ , then the distance between P and Q, denoted by | | P − Q | | , is 2 2 2 given by | | P − Q | | = (x1 − y1) + (x2 − y2) + … + (xn − yn). Prepared by GJ Villamin, 2024 ff fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.1. Distance). If P(x1, x2, …, xn) and Q(y1, y2, …, yn) are two n points in ℝ , then the distance between P and Q, denoted by | | P − Q | | , is 2 2 2 given by | | P − Q | | = (x1 − y1) + (x2 − y2) + … + (xn − yn). n (De nition 2.2. Open Ball). If A is a point in ℝ and r is a positive number, n then the open ball B(A, r) is de ned to be the set of all points P in ℝ such that | | P − A | | < r. Prepared by GJ Villamin, 2024 ff fi fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.1. Distance). If P(x1, x2, …, xn) and Q(y1, y2, …, yn) are two points in n ℝ , then the distance between P and Q, denoted by | | P − Q | | , is given by 2 2 2 | | P − Q | | = (x1 − y1) + (x2 − y2) + … + (xn − yn). n (De nition 2.2. Open Ball). If A is a point in ℝ and r is a positive number, then the n open ball B(A, r) is de ned to be the set of all points P in ℝ such that | | P − A | | < r. n (De nition 2.3. Closed Ball). If A is a point in ℝ and r is a positive number, then the n open ball B(A, r) is de ned to be the set of all points P in ℝ such that | | P − A | | ≤ r. Prepared by GJ Villamin, 2024 ff fi fi fi fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables n (De nition 2.1. Distance). If P(x1, x2, …, xn) and Q(y1, y2, …, yn) are two points in ℝ , then the distance between P and Q, denoted by | | P − Q | | , is given by 2 2 2 | | P − Q | | = (x1 − y1) + (x2 − y2) + … + (xn − yn). n (De nition 2.2. Open Ball). If A is a point in ℝ and r is a positive number, then the open n ball B(A, r) is de ned to be the set of all points P in ℝ such that | | P − A | | < r. n (De nition 2.3. Closed Ball). If A is a point in ℝ and r is a positive number, then the n open ball B(A, r) is de ned to be the set of all points P in ℝ such that | | P − A | | ≤ r. Open and closed balls are more commonly referred to as open and closed intervals in 1 2 3 ℝ , open and closed disks in ℝ , and open and closed spheres in ℝ. Prepared by GJ Villamin, 2024 ff fi fi fi fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.4. Limit of a Function of n Variables). Let f be a function of n variables which is de ned on some open ball B(A, r), except possibly at the point A itself. Then the limit of f(P) as P approaches A is L, written as lim f(P) = L, if for every ϵ > 0, there exists δ > 0 such that if P→A 0 < | | P − A | | < δ, then | f(P) − L | < ϵ. Prepared by GJ Villamin, 2024 ff fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.4. Limit of a Function of n Variables). Let f be a function of n variables which is de ned on some open ball B(A, r), except possibly at the point A itself. Then the limit of f(P) as P approaches A is L, written as lim f(P) = L, if for every ϵ > 0, there P→A exists δ > 0 such that if 0 < | | P − A | | < δ, then | f(P) − L | < ϵ. (De nition 2.5. Limit of a Function of Two Variables). Let f be a function of two variables which is de ned on some open disk B((x0, y0), r), except possibly at the point (x0, y0) itself. Then the limit of f(x, y) as (x, y) approaches (x0, y0) is L, written as lim f(x, y) = L, if for every ϵ > 0, there exists δ > 0 such that if (x,y)→(x0,y0) 2 2 0< (x − x0) + (y − y0) < δ, then | f(x, y) − L | < ϵ. Prepared by GJ Villamin, 2024 ff fi fi fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.6. Accumulation Point). A point P0 is said to be an n accumulation point of a set S of points in ℝ if every open ball B(P0, r) contains in nitely many points of S. Prepared by GJ Villamin, 2024 ff fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.6. Accumulation Point). A point P0 is said to be an n accumulation point of a set S of points in ℝ if every open ball B(P0, r) contains in nitely many points of S. Example: S = {x ∈ ℝ ∣ x > 0} Prepared by GJ Villamin, 2024 ff fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.6. Accumulation Point). A point P0 is said to be an n accumulation point of a set S of points in ℝ if every open ball B(P0, r) contains in nitely many points of S. Example: S = {x ∈ ℝ ∣ x > 0} x = 0 is an accumulation point of S. Prepared by GJ Villamin, 2024 ff fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.6. Accumulation Point). A point P0 is said to be an n accumulation point of a set S of points in ℝ if every open ball B(P0, r) contains in nitely many points of S. Example: S = {x ∈ ℝ ∣ x > 0} x = 0 is an accumulation point of S. x = 5 is an accumulation point of S. Prepared by GJ Villamin, 2024 ff fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.6. Accumulation Point). A point P0 is said to be an accumulation point n of a set S of points in ℝ if every open ball B(P0, r) contains in nitely many points of S. Example: S = {x ∈ ℝ ∣ x > 0} x = 0 is an accumulation point of S. x = 5 is an accumulation point of S. x = 20 is an accumulation point of S. Prepared by GJ Villamin, 2024 ff fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.6. Accumulation Point). A point P0 is said to be an accumulation point of a set n S of points in ℝ if every open ball B(P0, r) contains in nitely many points of S. Example: S = {x ∈ ℝ ∣ x > 0} x = 0 is an accumulation point of S. x = 5 is an accumulation point of S. x = 20 is an accumulation point of S. Any real number x in S is an accumulation point of S. Prepared by GJ Villamin, 2024 ff fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.6. Accumulation Point). A point P0 is said to be an n accumulation point of a set S of points in ℝ if every open ball B(P0, r) contains in nitely many points of S. Example: S = {(x, y) ∣ x ∈ ℤ, y ∈ ℤ} Prepared by GJ Villamin, 2024 ff fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.6. Accumulation Point). A point P0 is said to be an n accumulation point of a set S of points in ℝ if every open ball B(P0, r) contains in nitely many points of S. Example: S = {(x, y) ∣ x ∈ ℤ, y ∈ ℤ} S has no accumulation point. Prepared by GJ Villamin, 2024 ff fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.6. Accumulation Point). A point P0 is said to be an n accumulation point of a set S of points in ℝ if every open ball B(P0, r) contains in nitely many points of S. Example: {1 if n is even 0 if n is odd {an} where an = Prepared by GJ Villamin, 2024 ff fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.6. Accumulation Point). A point P0 is said to be an n accumulation point of a set S of points in ℝ if every open ball B(P0, r) contains in nitely many points of S. Example: {1 if n is even 0 if n is odd {an} where an = 0 and 1 are the accumulation points of {an}. Prepared by GJ Villamin, 2024 ff fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (De nition 2.6. Accumulation Point). A point P0 is said to be an n accumulation point of a set S of points in ℝ if every open ball B(P0, r) contains in nitely many points of S. (De nition 2.7. Limit of a Function of Two Variables). Let f be a function 2 de ned on a set of points S in ℝ , and let (x0, y0) be an accumulation point of S. Then the limit of f(x, y) as (x, y) approaches (x0, y0) in S is L, written as lim f(x, y) = L, if for every ϵ > 0, there exists δ > 0 such that if (x,y)→(x0,y0) 0 < | | (x, y) − (x0, y0) | | < δ, then | f(x, y) − L | < ϵ, and (x, y) is in S. Prepared by GJ Villamin, 2024 ff fi fi fi fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (Theorem 2.8. Limit Theorems). Let P = (x1, …, xi, …, xn) and A = (a1, …, ai, …, an). lim P→A c = c, for any constant c. lim P→A xi = ai lim x P→A i n = a n, for any positive integer n. i lim P→A n xi = n ai , for any positive integer n, provided ai > 0. lim P→A n xi = n ai , for any positive even integer n if ai ≤ 0. Prepared by GJ Villamin, 2024 ff  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (Theorem 2.8. Limit Theorems). lim f1(P) = L1, lim f2(P) = L2, …, lim fn(P) = Ln, then If P→A P→A P→A lim ( f1(P) ± f2(P) ± … ± fn(P)) = L1 ± L2 ± … ± Ln. P→A If lim f1(P) = L1, lim f2(P) = L2, …, lim fn(P) = Ln, then P→A P→A P→A lim ( f1(P) ⋅ f2(P) ⋅ … ⋅ fn(P)) = L1 ⋅ L2 ⋅ … ⋅ Ln. P→A f(P) L If lim f(P) = L and lim g(P) = M, then lim = , provided M ≠ 0. P→A P→A P→A g(P) M n If lim f(P) = L, then lim n f(P) = L , provided (1) L > 0 and n is any positive integer, or (2) L ≤ 0 P→A P→A and n is a positive odd integer. Prepared by GJ Villamin, 2024 ff  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (Theorem 2.8. Limit Theorems). Suppose that the functions f, g, and h are de ned on some open ball B containing A except possibly at A itself, and that f(P) ≤ g(P) ≤ h(P) for all P in B for which P ≠ A. Also suppose that lim f(P) and lim h(P) both P→A P→A exist and are equal to L. Then lim g(P) also exists and is equal to L. P→A Prepared by GJ Villamin, 2024 ff fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (Theorem 2.9.). Suppose that the function f is de ned for all points on an open disk having its center at (x0, y0), except possibly at (x0, y0) itself, and 2 lim f(x, y) = L. Then if S is any set of points in ℝ having (x0, y0) as (x,y)→(x0,y0) an accumulation point, lim f(x, y) exists and always has the value L. (x,y)→(x0,y0) Prepared by GJ Villamin, 2024 ff fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables (Theorem 2.9.). Suppose that the function f is de ned for all points on an open disk having its center at (x0, y0), except possibly at (x0, y0) itself, and 2 lim f(x, y) = L. Then if S is any set of points in ℝ having (x0, y0) as (x,y)→(x0,y0) an accumulation point, lim f(x, y) exists and always has the value L. (x,y)→(x0,y0) (Theorem 2.10.). If the function f has di erent limits as (x, y) approaches (x0, y0) through two distinct sets of points having (x0, y0) as an accumulation point, then lim f(x, y) does not exist. (x,y)→(x0,y0) Prepared by GJ Villamin, 2024 ff ff fi  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables xy Illustration: lim f(x, y) where f(x, y) = (x,y)→(0,0) x +y 2 2 dom f Prepared by GJ Villamin, 2024 ff  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables xy Illustration: lim f(x, y) where f(x, y) = (x,y)→(0,0) x +y 2 2 dom f = ℝ* × ℝ* Prepared by GJ Villamin, 2024 ff  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables xy Illustration: lim f(x, y) where f(x, y) = (x,y)→(0,0) x +y 2 2 dom f = ℝ* × ℝ* = {(x, y) ∈ ℝ ∣ (x, y) ≠ (0,0)} Prepared by GJ Villamin, 2024 ff  Limit of Functions of Several Variables Di erential Calculus of Functions of Several Variables xy Illustration: lim f(