Calculus II MTH301 Handouts PDF

Calculus II MTH301 Virtual University of Pakistan Knowledge beyond the boundaries 1 © Copyright Virtual University of Pakistan  Table of Contents Lecture No. 1...

Calculus II MTH301 Virtual University of Pakistan Knowledge beyond the boundaries 1 © Copyright Virtual University of Pakistan  Table of Contents Lecture No. 1 Introduction ……………………….………………………………...3 Lecture No. 2 Values of function ……………………… ……………………………..7 Lecture No. 3 Elements of three dimensional geometry ………….…………………..11 Lecture No. 4 Polar co-ordinates ………….…………………………………………..17 Lecture No. 5 Limits of multivariable function……...……………………………….24 Lecture No. 6 Geometry of continuous functions ……..…..…………………………..31 Lecture No. 7 Geometric meaning of partial deri.vative ….…………………………...38 Lecture No. 8 Euler theorm……………………...……………………………………. 42 Lecture No.9 Examples ………………………..……………………………………..48 Lecture No. 10 Introduction to vectors ………...………………..……………………..53 Lecture No. 11 The triple scalar product or Box product ………………….…………..62 Lecture No. 12 Tangent Planes to the surfaces ………………………………………69 Lecture No. 13 Noarmal Lines ….……...……………………………………………76 Lecture No. 14 Extrema of function of two variables……… ……………………..…..83 Lecture No. 15 Examples …….…………………………… …..……………………..87 Lecture No. 16 Extreme Valued Theorm ……..………… ………………..…………..93 Lecture No. 17 Example………………………………… ……………………………98 Lecture No. 18 Revision Of Integration …………… … …………………………..105 Lecture No. 19 Use of integrals……………………...………………………………..109 Lecture No. 20 Double integral for non-rectangular region ……………………..113 Lecture No. 21 Examples …………………...………………………………………..117 Lecture No. 22 Examples ……………..……………………………………………..121 Lecture No. 23 Polar co-ordinate systems ….………………………………………..124 Lecture No. 24 Sketching ………..…………………………………………………..128 Lecture No. 25 Double Integrals in Polar co-ordinates ….…………….………...…...133 Lecture No. 26 Examples …………..………………………………………….……..137 Lecture No. 27 Vector Valued Functions ………………………………………..…140 Lecture No. 28 Limits Of Vector Valued Functions ……………………………..144 Lecture No. 29 Change Of Parameter ………………….………… ……………….150 Lecture No. 30 Exact Differential …………………………………………………..155 Lecture No. 31 Line Intagral ………………………………………...……………...161 Lecture No. 32 Examples ………………………..…………………………………..166 Lecture No. 33 Examples …..………………………………………………………..170 Lecture No. 34 Examples ….………………………………………………………..174 Lecture No. 35 Definite...……………………………………..……………………..181 Lecture No. 36 Scalar Field……………………………………..……………………184 Lecture No. 37 Examples ………………………………………….………..……...188 Lecture No. 38 Vector Filed …………..……………………………………………..192 Lecture No. 39 Periodic Functions…………………………………………………..197 Lecture No. 40 Fourier Series ………..……………………………………………..201 Lecture No. 41 Examples ……………………………………………………..206 Lecture No. 42 Examples …………………………………………………………..212 Lecture No. 43 Functions with periods other than 2 π …………….………..…...217 Lecture No. 44 Laplace Transforms ………………………………………………..222 Lecture No. 45 Theorems…. ………………………………………………………..227 2 © Copyright Virtual University of Pakistan 1-Introduction VU Lecture No-1 Introduction – Calculus is the mathematical tool used to analyze changes in physical quantities. – Calculus is also Mathematics of Motion and Change. – Where there is motion or growth, where variable forces are at work producing acceleration, Calculus is right mathematics to apply. Differential Calculus Deals with the Problem of Finding (1)Rate of change. (2)Slope of curve. Velocities and acceleration of moving bodies. Firing angles that give cannons their maximum range. The times when planets would be closest together or farthest apart. Integral Calculus Deals with the Problem of determining a Function from information about its rates of Change. Integral Calculus Enables Us (1) To calculate lengths of curves. (2) To find areas of irregular regions in plane. (3) To find the volumes and masses of arbitrary solids (4) To calculate the future location of a body from its present position and knowledge of the forces acting on it. Reference Axis System Before giving the concept of Reference Axis System we recall you the concept of real line and locate some points on the real line as shown in the figure below, also remember that the real number system consist of both Rational and Irrational numbers that is we can write set of real numbers as union of rational and irrational numbers. Here in the above figure we have locate some of the rational as well as irrational numbers and also note that there are infinite real numbers between every two real numbers. Now if you are working in two dimensions then you know that we take the two mutually perpendicular lines and call the horizontal line as x-axis and vertical line as y-axis and where these lines cut we take that point as origin. Now any point on the x-axis will be denoted by an order pair whose first element which is also known as abscissa is a real number and other element of the order pair which is also known as ordinate will has 0 values. Similarly any point on the y-axis can be representing by an order pair. Some points are shown in the figure below. Also note that these lines divide the plane into four regions, First ,Second ,Third and Fourth quadrants respectively. We take the positive real numbers at the right side of the origin and negative to the left side, in the case of x-axis. Similarly for y-axis and also shown in the figure. 3 © Copyright Virtual University of Pakistan 1-Introduction VU Location of a point Now we will illustrate how to locate the point in the plane using x and y axis. Draw two perpendicular lines from the point whose position is to be determined. These lines will intersect at some point on the x-axis and y-axis and we can find out these points. Now the distance of the point of intersection of x-axis and perpendicular line from the origin is the X-C ordinate of the point P and similarly the distance from the origin to the point of intersection of y-axis and perpendicular line is the Y-coordinate of the point P as shown in the figure below. In space we have three mutually perpendicular lines as reference axis namely x ,y and z axis. Now you can see from the figure below that the planes x= 0 ,y=0 and z=0 divide the space into eight octants. Also note that in this case we have (0,0,0) as origin and any point in the space will have three coordinates. 4 © Copyright Virtual University of Pakistan 1-Introduction VU Sign of co-ordinates in different octants First of all note that the equation x=0 represents a plane in the 3d space and in this plane every point has its x-coordinate as 0, also that plane passes through the origin as shown in the figure above. Similarly y=0 and z=0 are also define a plane in 3d space and have properties similar to that of x=0.Such that these planes also pass through the origin and any point in the plane y=0 will have y-coordinate as 0 and any point in the plane z=0 has z-coordinate as 0. Also remember that when two planes intersect we get the equation of a line and when two lines intersect then we get a plane containing these two lines. Now note that by the intersection of the planes x=0 and z=0 we get the line which is our y-axis. Also by the intersection of x=0 and y=0 we get the line which is z-axis, similarly you can easily see that by the intersection of z=0 and y=0 we get line which is x-axis. Now these three planes divide the 3d space into eight octants depending on the positive and negative direction of axis. The octant in which every coordinate of any point has positive sign is known as first octant formed by the positive x, y and z –axis. Similarly in second octant every points has x-coordinate as negative and other two coordinates as positive correspond to negative x- axis and positive y and z axis. Now one octant is that in which every point has x and y coordinate negative and z- coordinate positive, which is known as the third octant. Similarly we have eight octants depending on the sign of coordinates of a point. These are summarized below. First octant (+, +, +) Formed by positive sides of the three axis. Second octant (-, +, + ) Formed by –ve x-axis and positive y and z-axis. Third octant ( -, -, +) Formed by –ve x and y axis with positive z-axis. Fourth octant ( +, -, +) Formed by +ve x and z axis and –ve y-axis. Fifth octant (+, +, -) Formed by +ve x and y axis with -ve z-axis. Sixth octant ( -, +, -) Formed by –ve x and z axis with positive y-axis. Seventh octant ( -, -, -) Formed by –ve sides of three axis. Eighth octant ( +, -, -) Formed by -ve y and z-axis with +ve x-axis. (Remember that we have two sides of any axis one of positive values and the other is of negative values) Now as we told you that in space we have three mutually perpendicular lines as reference axis. So far you are familiar with the reference axis for 2d which consist of two perpendicular lines namely x-axis and y-axis. For the reference axis of 3d space we need another perpendicular axis which can be obtained by the cross product of the two vectors, now the direction of that vector can be obtained by Right hand rule. This is illustartaed below with diagram. 5 © Copyright Virtual University of Pakistan 1-Introduction VU Concept of a Function Historically, the term, function, denotes the dependence of one quantity on other quantity. The quantity x is called the independent variable and the quantity y is called the dependent variable. We write y = f (x) and we read y is a function of x. The equation y = 2x defines y as a function of x because each value assigned to x determines unique value of y. Examples of function – The area of a circle depends on its radius r by the equation A= πr2 so, we say that A is a function of r. – The volume of a cube depends on the length of its side x by the equation V= x3 so, we say that V is a function of x. – The velocity V of a ball falling freely in the earth’s gravitational field increases with time t until it hits the ground, so we say that V is function of t. – In a bacteria culture, the number n of present after one day of growth depends on the number N of bacteria present initially, so we say that N is function of n. Function of Several Variables Many functions depend on more than one independent variable. Examples The area of a rectangle depends on its length l and width w by the equation A = l w , so we say that A is a function of l and w. The volume of a rectangular box depends on the length l, width w and height h by the equation V = l w h so, we say that V is a function of l , w and h. The area of a triangle depends on its base length l and height h by the equation A= ½ l h, so we say that A is a function of l and h. The volume V of a right circular cylinder depends on its radius r and height h by the equation V= πr2h so, we say that V is a function of r and h. Home Assignments: In the first Lecture we recall some basic terminologies which are essential and prerequisite for this course. You can find the Home Assignments on the last page of Lecture # 1 at LMS. 6 © Copyright Virtual University of Pakistan 2-Values of functions VU Lecture No-2 Values of functions: Consider the function f(x) = 2x2 –1, then f(1) = 2(1)2 –1 = 1, f(4) = 2(4)2 –1 = 31, f(-2) = 2(-2)2 –1 = 7 2 2 f(t-4) = 2(t-4) –1= 2t -16t + 31 These are the values of the function at some points. Example Now we will consider a function of two variables, so consider the function f(x,y) =x2y+1 then f(2,1) =(22)1+1=5, f(1,2) =(12)2+1=3, f(0,0) =(02)0+1=1, f(1,-3) =(12)(-3)+1=-2, f(3a,a) =(3a)2a+1=9a3+1, f(ab,a-b) =(ab)2(a-b)+1=a3b2-a2b3+1 These are values of the function at some points. Example: Now consider the function f ( x, y ) = x + 3 xy then (a) f (2, 4) = 2 + 3 (2)(4) = 2 + 3 8 = 2+2 = 4 (b) f (t , t 2 ) = t + 3 (t )(t 2 ) = t + 3 t 3 = t + t = 2t (c) f ( x, x 2 ) = x + 3 ( x )( x 2 ) = x + 3 x3 = x + x = 2 x (d) f (2 y 2 , 4 y ) = 2 y 2 + 3 (2 y 2 )(4 y ) = 2 y 2 + 3 8 y 3 = 2 y 2 + 2 y Example: Now again we take another function of three variables f ( x, y, z ) = 1 − x 2 − y 2 − z 2 Then 1 1 1 1 1 f (0, , ) = 1 − 0 − ( )2 − ( )2 = 2 2 2 2 2 Example: Consider the function f(x,y,z) =xy2z3+3 then at certain points we have f(2,1,2) =(2)(1)2(2)3+3=19, f(0,0,0) =(0)(0)2(0)3+3=3, f(a,a,a) =(a)(a)2(a)3+3=a6+3 f(t,t2,-t) =(t)(t2)2(-t)3+3=-t8+3, f(-3,1,1) =(-3)(1)2(1)3+3=0 Example: Consider the function f(x,y,z) =x2y2z4 where x(t) =t3 y(t)= t2and z(t)=t (a) f(x(t),y(t),z(t)) =[x(t)]2[y(t)]2[z(t)]4=[ t3]2[t2]2[t]4= t14 (b) f(x(0),y(0),z(0)) =[x(0)]2[y(0)]2[z(0)]4=[ 03]224= 0 Example: Let us consider the function f(x,y,z) = xyz + x then f(xy,y/x,xz) = (xy)(y/x)(xz) + xy = xy2z+xy. Example: Let us consider g(x,y,z) =z Sin(xy), u(x,y,z) =x2z3 , v(x,y,z) =Pxyz, xy w( x, y, z ) = Then. z g(u(x,y,z), v(x,y,z), w(x,y,z)) = w(x,y,z) Sin(u(x,y,z) v(x,y,z)) Now by putting the values of these functions from the above equations we get xy xy g(u(x,y,z), v(x,y,z), w(x,y,z)) = Sin[(x2z3)( Pxyz)] = Sin[(Pyx3z4)]. z z 7 © Copyright Virtual University of Pakistan 2-Values of functions VU Example: Consider the function g(x,y) =y Sin(x2y) and u(x,y) =x2y3 v(x,y) = π xy Then g(u(x,y), v(x,y)) = v(x,y) Sin([u(x,y) ]2 v(x,y)) By putting the values of these functions we get g(u(x,y), v(x,y)) = π xy Sin([x2y3]2 π xy) = π xy Sin(x5y7 ). Function of One Variable A function f of one real variable x is a rule that assigns a unique real number f( x ) to each point x in some set D of the real line. Function of two Variables A function f in two real variables x and y, is a rule that assigns unique real number f (x,y) to each point (x,y) in some set D of the xy-plane. Function of three variables: A function f in three real variables x, y and z, is a rule that assigns a unique real number f (x,y,z) to each point (x,y,z) in some set D of three dimensional space. Function of n variables: A function f in n variable real variables x ,x , x ,……, x , is a rule that assigns a unique 1 2 3 n real number w = f(x1, x2, x3,……, xn) to each point (x1, x2, x3,……, xn) I n some set D of n dimensional space. Circles and Disks: PARABOLA Parabola y = -x2 8 © Copyright Virtual University of Pakistan 2-Values of functions VU General equation of the Parabola opening upward or downward is of the form y = f(x) = ax2+bx + c. Opening upward if a > 0. Opening downward if a < 0. x co-ordinate of the vertex is given by x0 = -b/2a. So the y co-ordinate of the vertex is y0= f(x0) axis of symmetry is x = x0. As you can see from the figure below Sketching of the graph of parabola y = ax2+bx + c Finding vertex: x – co-ordinate of the vertex is given by x0= - b/2a So, y – co-ordinate of the vertex is y0= a x02+b x0 + c. Hence vertex is V(x0 , y0). Example: Sketch the parabola y = - x2 + 4x Solution: Since a = -1 < 0, parabola is opening downward. Vertex occurs at x = - b/2a = (-4)/2(-1) =2. Axis of symmetry is the vertical line x = 2. The y-co-ordinate of the vertex isy = -(2)2 + 4(2) = 4. Hence vertex is V(2 , 4 ). The zeros of the parabola (i.e. the point where the parabola meets x-axis) are the solutions to -x2 +4x = 0 so x = 0 and x = 4. Therefore (0,0)and (4,0) lie on the parabola. Also (1,3) and (3,3) lie on the parabola. Graph of y = - x2 + 4x Example y = x2 - 4x+3 Solution: Since a = 1 > 0, parabola is opening upward.Vertex occurs at x = - b/2a = (4)/2 =2.Axis of symmetry is the vertical line x = 2. The y co-ordinate of the vertex is y = (2)2 - 4(2) + 3 = -1.Hence vertex is V(2 , -1 )The zeros of the parabola (i.e. the point where the parabola meets x-axis) are the solutions to x2 - 4x + 3 = 0, so x = 1 and x = 3.Therefore (1,0)and (3,0) lie on the parabola. Also (0 ,3 ) and (4, 3 ) lie on the parabola. 9 © Copyright Virtual University of Pakistan 2-Values of functions VU Graph of y = x2 - 4x+3 Ellipse Hyperbola Home Assignments: In this lecture we recall some basic geometrical concepts which are prerequisite for this course and you can find all these concepts in the chapter # 12 of your book Calculus By Howard Anton. 10 © Copyright Virtual University of Pakistan  3-Elements of three dimensional geometry VU Lecture No-3 Elements of three dimensional geometry Distance formula in three dimension Let P ( x1 , y1 , z1 ) and Q ( x2 , y2 , z2 ) be two points such that PQ is not parallel to one of the coordinate axis Then PQ = ( x2 − x1 )2 + ( y2 − y1 ) 2 + ( z2 − z1 )2 Which is known as Distance fromula between the points P and Q. Example of distance formula Let us considerthe points A (3, 2, 4), B (6, 10, −1), and C (9, 4, 1) Then | AB | = (6 − 3)2 + (10 − 2)2 + (− 1 − 4)2 = 98 = 7 2 2 2 2 | AC | = (9 − 3) + (4 − 2) + (1 − 4) = 49 = 7 | BC | = (9 − 6)2 + (4 − 10)2 + (1 + 1)2 = 49 = 7 Mid point of two points If R is the middle point of the line segment PQ, then the co-ordinates of the middle points are x= (x1+x2)/2 , y= (y1+y2)/2 , z= (z1+z2)/2 Let us consider tow points A(3,2,4) and B(6,10,-1) Then the co-ordinates of mid point of AB is [(3+6)/2,(2+10)/2,(4-1)/2] = (9/2,6,3/2) Direction Angles The direction angles α β , γ of a lineare defined as α = Angle between lineand the positive x-axis β = Angle between line and the positivey-axis γ = Angle between lineand the positive z-axis. By definition, each of these angles lies between 0. and π Direction Ratios Cosines of direction angles are called direction cosines Any multiple of direction cosines are called direction numbers or direction ratios of the line L. Given a point, finding its Direction cosines y-axis 11 © Copyright Virtual University of Pakistan 3-Elements of three dimensional geometry VU P(x,y) From triangle we can β write r y cos α = x/r cos β = y/r α O x x-axis Direction angles of a Line The angles which a line makes with positive x,y and z-axis are known as Direction Angles. In the above figure the blue line has direction angles as α, and which are the angles which blue line makes with x,y and z-axis respectively. Direction cosines: Now if we take the cosine of the Direction Angles of a line then we get the Direction cosines of that line. So the Direction Cosines of the above line are given by x x cos α = OP = x2 +y2 +z2 y y cos β = OP = x2 +y2 +z2 Similarly, z z cos γ = OP = x2 +y2 +z2 c os 2 α + c os 2 β + cos 2 γ = 1. Direction cosines and direction ratios of a line joining two points For a line joining two points P(x1, y1, z1) and Q(x2, y2, x2) the direction ratios are 12 © Copyright Virtual University of Pakistan 3-Elements of three dimensional geometry VU x 2 - x1 y 2 - y1 z -z x2 - x1, y2 - y1, z2 - z1 and the directions cosines are , and 2 1. PQ PQ PQ Example For a line joining two points P(1,3,2) and Q(7,-2,3) the direction ratios are 7 - 1, -2 – 3 , 3 – 2 6 , -5 , 1 and the directions cosines are 6/√62 , -5/√62 , 1/ √62 In two dimensional space the graph of an equation relating the variables x and y is the set of all point (x, y) whose co-ordinates satisfy the equation. Usually, such graphs are curves. In three dimensional space the graph of an equation relating the variables x, y and z is the set of all point (x, y, z) whose co-ordinates satisfy the equation. Usually, such graphs are surfaces. Intersection of two surfaces Intersection of two surfaces is a curve in three dimensional space. It is the reason that a curve in three dimensional space is represented by two equations representing the intersecting surfaces. Intersection of Cone and Sphere Intersection of Two Planes If the two planes are not parallel, then they intersect and their intersection is a straight line. Thus, two non-parallel planes represent a straight line given by two simultaneous linear equations in x, y and z and are known as non-symmetric form of equations of a straight line. 13 © Copyright Virtual University of Pakistan 3-Elements of three dimensional geometry VU REGION DESCRIPTION EQUATION xy-plane Consists of all points of the form (x, y, 0) z=0 xz-plane Consists of all points of the form (x, 0, z) y=0 yz-plane Consists of all points of the form (0, y, z) x=0 x-axis Consists of all points of the form (x, 0, 0) y = 0, z = 0 y-axis Consists of all points of the form (0, y, 0) z = 0, x = 0 z-axis Consists of all points of the form (0, 0, z) x = 0, y = 0 Planes parallel to Co-ordinate Planes General Equation of Plane Any equation of the form ax + by + cz + d = 0 where a, b, c, d are real numbers,represent a plane. Sphere 14 © Copyright Virtual University of Pakistan 3-Elements of three dimensional geometry VU Right Circular Cone Horizontal Circular Cylinder 15 © Copyright Virtual University of Pakistan 3-Elements of three dimensional geometry VU Horizontal Elliptic Cylinder Overview of Lecture # 3 Chapter # 14 Three Diamentional Space Page # 657 Book CALCULUS by HOWARD ANTON 16 © Copyright Virtual University of Pakistan 4-Polar Coordinates VU Lecture -4 Polar co-ordinates You know that position of any point in the plane can be obtained by the two perpendicular lines known as x and y axis and together we call it as Cartesian coordinates for plane. Beside this coordinate system we have another coordinate system which can also use for obtaining the position of any point in the plane. In that coordinate system we represent position of each particle in the plane by “r” and “θ ”where “r” is the distance from a fixed point known as pole and θ is the measure of the angle. P (r, θ) O θ Initial ray “O” is known as pole. Conversion formula from polar to Cartesian coordinates and vice versa P(x, y) =P(r, θ ) r y θ x From above diagram and remembering the trigonometric ratios we can write x = r cos θ, y = r sin θ. Now squaring these two equations and adding we get, x2 + y2 = r2 Dividing these equations we get y/x = tanθ These two equations gives the relation between the Plane polar and Plane Cartesian coordinates. Rectangular co-ordinates for 3d Since you know that the position of any point in the 3d can be obtained by the three mutually perpendicular lines known as x ,y and z – axis and also shown in figure below, these coordinate axis are known as Rectangular coordinate system. 17 © Copyright Virtual University of Pakistan 4-Polar Coordinates VU Cylindrical co-ordinates Beside the Rectangular coordinate system we have another coordinate system which is used for getting the position of the any particle is in space known as the cylindrical coordinate system as shown in the figure below. Spherical co-ordinates Beside the Rectangular and Cylindrical coordinate systems we have another coordinate system which is used for getting the position of the any particle is in space known as the spherical coordinate system as shown in the figure below. 18 © Copyright Virtual University of Pakistan 4-Polar Coordinates VU Conversion formulas between rectangular and cylindrical co-ordinates Now we will find out the relation between the Rectangular coordinate system and Cylindrical coordinates. For this consider any point in the space and consider the position of this point in both the axis as shown in the figure below. In the figure we have the projection of the point P in the xy-Plane and write its position in plane polar coordinates and also represent the angle θ now from that projection we draw perpendicular to both of the axis and using the trigonometric ratios find out the following relations. (r, θ, z) →(x, y, z) x = rcosθ, y = rsinθ, z=z y r= x2 +y2 , tanθ = , z=z x Conversion formulas between cylindrical and spherical co-ordinates Now we will find out the relation between spherical coordinate system and Cylindrical coordinate system. For this consider any point in the space and consider the position of this point in both the axis as shown in the figure below. First we will find the relation between Planes polar to spherical, from the above figure you can easily see that from the two right angled triangles we have the following relations. (ρ ,θ,φ ) → (r, θ, z) 19 © Copyright Virtual University of Pakistan 4-Polar Coordinates VU r = ρ sin φ , θ = θ, z = ρ cos φ Now from these equations we will solve the first and second equation for ρ and φ. Thus we have (r, θ, z) → (ρ ,θ,φ ) r ρ= r2 +z 2 θ = θ, tan φ = z Conversion formulas between rectangular and spherical co-ordinates (ρ, θ, Φ) → (x, y, z) Since we know that the relation between Cartesian coordinates and Polar coordinates are x = r cos θ, y = r sin θ and z = z.We also know the relation between Spherical and cylindrical coordinates are, r = ρ sin φ , θ = θ, z = ρ cos φ Now putting this value of “r” and “z” in the above formulas we get the relation between spherical coordinate system and Cartesian coordinate system. Now we will find ( x, y, z) → ( ρ, θ, Φ) 2 2 2 2 2 2 x + y +z = (ρsin Φ cos θ) + (ρsin Φ sin θ) + (ρ cos Φ) 2 2 2 2 2 = ρ {sin Φ(cos θ + sin θ) +cos Φ)} 2 2 2 2 2 = ρ (sin Φ + cos Φ) = ρ ρ= x2 + y2 +z2 z Tanθ = y/x and Cos Φ = x2 + y2 +z2 Constant surfaces in rectangular co-ordinates The surfaces represented by equations of the form x = x0, y = y0, z = z0 where xo, yo, zo are constants, are planes parallel to the xy-plane, xz-plane and xy- plane, respectively. Also shown in the figure 20 © Copyright Virtual University of Pakistan 4-Polar Coordinates VU Constant surfaces in cylindrical co-ordinates The surface r = ro is a right cylinder of radius ro centered on the z-axis. At each point (r, θ, z) this surface on this cylinder, r has the value r0 , z is unrestricted and 0 ≤ θ < 2π. The surface θ = θ0 is a half plane attached along the z-axis and making angle θ0 with the positive x-axis. At each point (r, θ, z) on the surface ,θ has the value θ0, z is unrestricted and r ≥0. The surfaces z = zo is a horizontal plane. At each point (r, θ, z) this surface z has the value z0 , but r and θ are unrestricted as shown in the figure below. Constant surfaces in spherical co-ordinates The surface ρ = ρo consists of all points whose distance ρ from origin is ρo. Assuming that ρo to be nonnegative, this is a sphere of radius ρo centered at the origin. The surface θ = θ0 is a half plane attached along the z-axis and making angle θ0 with the positive x- axis. The surface Φ = Φ0 consists of all points from which a line segment to the origin makes an angle of Φ0 with the positive z-axis. Depending on whether 0< Φ0 < π/2 or π/2 < Φ0< π, this will be a cone opening up or opening down. If Φ0 = π/2, then the cone is flat and the surface is the xy-plane. 21 © Copyright Virtual University of Pakistan 4-Polar Coordinates VU Spherical Co-ordinates in Navigation Spherical co-ordinates are related to longitude and latitude coordinates used in navigation. Let us consider a right handed rectangular coordinate system with origin at earth’s center, positive z-axis passing through the north pole,and x-axis passing through the prime meridian. Considering earth to be a perfect sphere of radius ρ = 4000 miles, then each point has spherical coordinates of the form (4000,θ,Φ) where Φ and θ determine the latitude and longitude of the point. Longitude is specified in degree east or west of the prime meridian and latitudes is specified in degree north or south of the equator. Domain of the Function In the above definitions the set D is the domain of the function. The Set of all values which the function assigns for every element of the domain is called the Range of the function. When the range consist of real numbers the functions are called the real valued function. NATURAL DOMAIN Natural domain consists of all points at which the formula has no divisions by zero and produces only real numbers. Examples Consider the Function ϖ = y − x 2. Then the domain of the function is y ≥ x 2 Which can be shown in the plane as 22 © Copyright Virtual University of Pakistan 4-Polar Coordinates VU and Range of the function is [ 0, ∞ ). Domain of function w = 1/xy is the whole xy- plane Excluding x-axis and y-axis, because at x and y axis all the points has x and y coordinates as 0 and thus the defining formula for the function gives us 1/0. So we exclude them. 23 © Copyright Virtual University of Pakistan 5-Limits of multivariable function VU Lecture No-5 Limit of Multivariable Function f (x,y)= sin -1(x+y) Domain of f is the region in which -1 ≤ x +y ≤ 1 y-axis x = -1 x =1 -1 ≤ x +y ≤ 1 y =1 x-axis y = -1 Domains and Ranges Functions Domain Range Entire space [0, ∞ ) ω = 2 x + 2 y+ z 2 1 ω = (x,y,z) ≠ (0, 0, 0) (0, ∞ ) x2 + y 2 + z2 Half space z>0 ( − ∞ ,∞ ) ω = xy lnz Examples of domain of a function f(x, y) = xy y - 1 Domain of f consists of region in xy plane where y ≥ 1 f (x, y) = x 2 + y2 - 4 Domain of f consists of region in xy plane where x 2 + y 2≥ 4 As shown in the figure 24 © Copyright Virtual University of Pakistan  5-Limits of multivariable function VU f(x, y) = lnxy Domain of f consists of region lying in first and third quadrants in xy plane as shown in above figure right side. xyz f(x, y,z) =e Domain of f consists of region of three dimensional space 4 - x2 f(x, y) = 2 y +3 Domain of f consists of region in xy plane x 2 ≤ 4 ,- 2 ≤ x ≤2 x=2 x = -2 y 2 2- 2 f(x, y, z) = 25- x - y - z Domain of f consists of region in three dimensional space occupied by sphere centre at (0, 0, 0) and radius 5. x 3 + 2 x 2y - x y - 2 y 2 f (x , y ) = x + 2y f(0,0) is not defined but we see that limit exits. 25 © Copyright Virtual University of Pakistan 5-Limits of multivariable function VU Approaching to (0,0) Approaching to (0,0) through f (x,y) through f (x,y) x-axis y-axis (0.5,0) 0.25 (0,0.1) -0.1 (0.25,0) 0.0625 (0,0.001) -0.001 (0.1,0) 0.01 (0,0.00001) 0.00001 (-0.25,0) 0.0625 (0,-0.001) 0.001 (-0.1,0) 0.01 (0,-0.00001) 0.00001 Approaching to (0,0) through f (x,y) y=x (0.5,0.5) -0.25 (0.1,0.1) -0.09 (0.01,0.01) -0.0099 (-0.5,-0.5) 0.75 (-0.1,-0.1) 0.11 (-0.01,-0.01) 0.0101 26 © Copyright Virtual University of Pakistan 5-Limits of multivariable function VU Example f (x,y) = 2xy 2 x +y f(0,0) is not defined and we see that limit also does not exit. Approaching to Approaching to (0,0) through f (x,y) (0,0) through f (x,y) x-axis (y = 0) y=x ( 0.5,0 ) 0 ( 0.5,0.5 ) 0.5 ( 0.1,0 ) 0 ( 0.25,0.25 ) 0.5 ( 0.01,0 ) 0 ( 0.1,0.1 ) 0.5 ( 0.001,0 ) 0 ( 0.05,0.05 ) 0.5 ( 0.0001,0 ) 0 ( 0.001,0.001 ) 0.5 ( -0.5,0 ) 0 ( -0.5,-0.5 ) 0.5 ( -0.1,0 ) 0 ( -0.25,-0.25 ) 0.5 ( -0.01,0 ) 0 ( -0.1,-0.1 ) 0.5 ( -0.001,0 ) 0 ( -0.05,-0.05 ) 0.5 ( -0.0001,0 ) 0 ( -0.001,-0.001 ) 0.5 xy lim 2 = 0 (along y = 0) (x,y) → (0,0) x + y2 xy lim 2 = 0.5 (along y = x) (x,y ) → (0,0) x + y2 xy lim does not exist. (x,y ) → (0,0) 2 x + y2 Example xy lim (x , y) → (0, 0) x + y2 2 Let (x , y) approach (0, 0) along the line y = x. xy x.x 1 f (x , y) = 2 2 = 2 2 = x ≠ 0. x +y x +x 1+1 1 = 2 27 © Copyright Virtual University of Pakistan 5-Limits of multivariable function VU Let (x , y) approach (0, 0) along the line y = 0. x. (0) f (x , y) = 2 2 = 0 , x ≠ 0. x + (0) Thus f(x, y) assumes two different values as (x,y) approaches (0,0) along two different paths. lim f (x, y) does not exist. (x, y) → (0, 0) We can approach a point in space through infinite paths some of them are shown in the y figure below. (x0,y0) (xy) x Rule for Non-Existence of a Limit If in Lim f(x , y) ( x , y) → ( a , b ) We get two or more different values as we approach (a, b) along different paths, then Lim f(x, y) (x, y) → (a, b) does not exist. The paths along which (a, b) is approached may be straight lines or plane curves through (a, b). Example x3 + x2 y − x − y2 Lim (x, y) → (2 , 1 ) x+ 2y Lim (x 3 + 2 x 2 y − xy − 2 y 2 ) ( x , y ) → (2 , 1 ) = Lim (x + 2 y ) ( x, y ) → (2 , 1 ) 28 © Copyright Virtual University of Pakistan  5-Limits of multivariable function VU Lim (x 3 + 2 x 2 y − x − y2 ) ( x, y) → ( 2 , 1 ) = Lim (x + 2 y ) ( x, y) → (2 , 1 ) Example xy Lim (x , y) → (0,0 ,0 ) x 2 + y2 We set x = r cos θ , y = r sin θ x r cos θ. r sin θ then 2 = x + y2 r cos2 θ + r2 sin2 θ 2 = r cos θ sinθ , for r >0 Since r = x 2 + y2 , r → 0 as ( x , y ) → (0, 0), x Lim = Lim r cosθ sinθ = 0, (x, y) → (0, 0) x2 + y2 r → 0 since | cos θ sin θ | ≤ 1 for all value of θ. RULES FOR LIMIT If lim f ( x, y ) = L1 and lim g ( x, y ) = L2 Then ( x , y ) →( x0 , y0 ) ( x , y ) →( x0 , y0 ) (a) lim cf ( x, y ) = cL1 (if c is constant) ( x , y ) →( x0 , y0 ) (b) lim { f ( x, y ) + g ( x, y )} = L1 + L2 ( x , y ) →( x0 , y0 ) lim { f ( x, y ) − g ( x, y )} = L1 − L2 ( x , y ) →( x0 , y0 ) (d) lim { f ( x, y ) g ( x, y )} = L1 L2 ( x , y ) →( x0 , y0 ) f ( x, y ) L1 (e) lim = (if L2 = 0) ( x , y ) → ( x0 , y0 ) g ( x, y ) L2 lim c = c (c a constant), lim x0 = x0 , lim y0 = y0 ( x , y ) →( x0 , y0 ) ( x , y ) →( x0 , y0 ) ( x , y ) →( x0 , y0 ) Similarly for the function of three variables. 29 © Copyright Virtual University of Pakistan 5-Limits of multivariable function VU Overview of lecture# 5 In this lecture we recall you all the limit concept which are prerequisite for this course and you can find all these concepts in the chapter # 16 (topic # 16.2)of your Calculus By Howard Anton. 30 © Copyright Virtual University of Pakistan 6-Geometry of continuous functions VU Lecture No -6 Geometry of continuous functions Geometry of continuous functions in one variable or Informal definition of continuity of function of one variable. A function is continuous if we draw its graph by a pen then the pen is not raised so that there is no gap in the graph of the function Geometry of continuous functions in two variables or Informal definition of continuity of function of two variables. The graph of a continuous function of two variables to be constructed from a thin sheet of clay that has been hollowed and pinched into peaks and valleys without creating tears or pinholes. Continuity of functions of two variables A function f of two variables is called continuous at the point (x0,y0) if 1. f (x0,y0) if defined. 2. lim f ( x, y ) exists. ( x , y )→( x0 , y0 ) 3. lim f ( x, y ) = f (x0,y0). ( x , y ) →( x0 , y0 ) The requirement that f (x0,y0) must be defined at the point (x0,y0) eliminates the possibility of a hole in the surface z = f (x0,y0) above the point (x0,y0). Justification of three points involving in the definition of continuity. (1) Consider the function of two variables x 2 + y 2 ln( x 2 + y 2 ) now as we know that the Log function is not defined at 0, it means that when x = 0 and y = 0 our function x 2 + y 2 ln( x 2 + y 2 ) is not defined. Consequently the surface z = x 2 + y 2 ln( x 2 + y 2 ) will have a hole just above the point (0,0)as shown in the graph of x 2 + y 2 ln( x 2 + y 2 ) (2) The requirement that lim f ( x, y ) exists ensures us that the surface z = f(x,y) of ( x , y )→( x0 , y0 ) the function f(x,y) doesn’t become infinite at (x0,y0) or doesn’t oscillate widely. 31 © Copyright Virtual University of Pakistan 6-Geometry of continuous functions VU 1 Consider the function of two variables now as we know that the Natural domain x2 + y 2 of the function is whole the plane except origin. Because at origin we have x = 0 and y =0 in the defining formula of the function we will have at that point 1/0 which is infinity. 1 Thus the limit of the function does not exists at origin. Consequently the x2 + y 2 1 surface z = will approaches towards infinity when we approaches towards x + y2 2 origin as shown in the figure above. (3) The requirement that lim f ( x, y ) = f (x0,y0) ( x , y ) →( x0 , y0 ) ensures us that the surface z = f(x,y) of the function f(x,y) doesn’t have a vertical jump or step above the point (x0,y0). Consider the function of two variables ⎧0 if x ≥ 0and y ≥ 0 f ( x, y ) = ⎨ ⎩1 otherwise now as we know that the Natural domain of the function is whole the plane. But you should note that the function has one value “0” for all the points in the plane for which both x and y have nonnegative values. And value “1” for all other points in the plane. Consequently the surface ⎧0 if x ≥ 0and y ≥ 0 z = f ( x, y ) = ⎨ has a jump as shown in the figure ⎩1 otherwise 32 © Copyright Virtual University of Pakistan 6-Geometry of continuous functions VU Example Check whether the limit exists or not for the function x2 lim f ( x, y ) = 2 ( x , y ) →(0,0) x + y2 Solution: First we will calculate the Limit of the function along x-axis and we get x2 lim f ( x, y ) = 2 = 1 (Along x-axis) ( x , y ) →(0,0) x +0 Now we will find out the limit of the function along y-axis and we note that the limit y2 is lim f ( x, y ) = 2 = 1 (Along y-axis). Now we will find out the limit of the ( x , y ) →(0,0) y +0 x2 1 function along the line y = x and we note that lim f ( x, y ) = 2 = (Along y = x) ( x , y ) →(0,0) x +x 2 2 It means that limit of the function at (0,0) doesn’t exists because it has different values along different paths. Thus the function cannot be continuous at (0,0). And also note that the function id not defined at (0,0) and hence it doesn’t satisfy two conditions of the continuity. Example Check the continuity of the function at (0,0) ⎧ sin( x2 + y 2 ) ⎪ if ( x, y) ≠ ( 0 , 0 ) f ( x , y) = ⎨ x2 + y2 ⎪⎩ 1 if ( x, y) = ( 0, 0 ) Solution: First we will note that the function is defined on the point where we have to check the Continuity that is the function has value at (0,0). Next we will find out the sin x Limit of the function at (0,0) and in evaluating this limit we use the result lim =1 x →0 x and note that 33 © Copyright Virtual University of Pakistan 6-Geometry of continuous functions VU Sin(x2 + y2 ) lim f(x,y) = lim x2 + y2 (x,y ) → (0,0) (x,y ) → (0,0) =1 = f(0, 0) This shows that f is continuous at (0,0) CONTINUITY OF FUNCTION OF THREE VARIABLES A function f of three variables is called continuous at a point (x0,y0,z0) if 1. f (x0,y0,z0) if defined. 2. lim f ( x, y, z ) exists; ( x , y , z ) →( x0 , y0 , z0 ) 3 lim f ( x, y, z ) = f(x0,y0,z0). ( x , y , z ) →( x0 , y0 , z0 ) EXAMPLE Check the continuity of the function y +1 f ( x, y , z ) = 2 x + y2 −1 Solution: First of all note that the given function is not defined on the cylinder x + y − 1 = 0.Thus the function is not continuous on the cylinder x 2 + y 2 − 1 = 0 2 2 And continuous at all other points of its domain. RULES FOR CONTINOUS FUNCTIONS (a) If g and h are continuous functions of one variable, then f(x, y) = g(x)h(y) is a continuous function of x and y (b) If g is a continuous function of one variable and h is a continuous function of two variables, then their composition f(x, y) = g(h(x,y)) is a continuous function of x and y. A composition of continuous functions is continuous. A sum, difference, or product of continuous functions is continuous. A quotient of continuous function is continuous, expect where the denominator is zero. EXAMPLE OF PRODUCT OF FUNCTIONS TO BE CONTINUED In general, any function of the form f(x, y) = Axmyn (m and n non negative integers) is continuous because it is the product of the continuous functions Axm and yn. The functio n f(x, y) = 3x2 y5 is continuous because it is the product of the continuous functions g(x) = 3x2 and h(y) = y5. CONTINUOUS EVERYWHERE A function f that is continuous at each point of a region R in 2-dimensional space or 3-dimensional space is said to be continuous on R. A function that is continuous at every point in 2-dimensional space or 3-dimensional space is called continuous everywhere or simply continuous. 34 © Copyright Virtual University of Pakistan 6-Geometry of continuous functions VU Example: (1)f (x,y) = ln(2x – y +1) The function f is continuous in the whole region where 2x > y-1, y < 2x+1.And its region is shown in figure below. y < 2x+1 1− xy (2) f ( x, y ) = e The function f is continuous in the whole region of xy-plane. −1 (3) f ( x, y ) = tan ( y − x ) The function f is continuous in the whole region of xy- plane. (4) f ( x, y ) = y − x The function is continuous where x ≥ y y x≥y Partial derivative Let f be a function of x and y. If we hold y constant, say y = y and view x as a variable, 0 then f(x, y ) is a function of x-alone. If this function is differentiable at 0 x = xo, then the value of this derivative is denoted by f (x , y ) and is called the Partial x o 0 derivative of f with respect of x at the point (x , y ). o 0 Similarly, if we hold x constant, say x = x and view y as a variable, then f (x , y ) is a 0 0 function of y alone. If this function is differentiable at y = y0, then the value of this derivative is denoted by f (x , y ) and is called the Partial derivative of f with respect y 0 0 of y at the point (x , y ) 0 0 35 © Copyright Virtual University of Pakistan 6-Geometry of continuous functions VU Example f ( x, y ) = 2 x 3 y 2 + 2 y + 4 x Trea ting y as a constant and differentiating with respect to x, we obtain fx (x, y) = 6x2y2 + 4 Trea ting x as a constant a nd differentiating with respect to y, we obtain fy (x, y) = 4x3y + 2 Substituting x = 1 and y = 2 in these partial-derivative formulas yields. fx (1, 2) = 6(1)2(2)2 + 4 = 28 fy (1, 2) = 4(1)3(2) + 2 = 10 Example Z = 4x2 - 2y + 7x4y5 ∂z = 8 x + 28 x 3 y 5 ∂x ∂z = −2 + 35 x 4 y 4 ∂y Example z = f ( x, y ) = x 2 sin 2 y Then to find the derivative of f with respect to x we treat y as a constant therefore ∂z = f x = 2 x sin 2 y ∂x Then to find the derivative of f with respect to y we treat x as a constant therefore ∂z = f y = x 2 2sin y cos y ∂y = x 2 sin 2 y Example ⎛ x2 + y 2 ⎞ z = ln ⎜ ⎟ ⎝ x+ y ⎠ By using the properties of the ln we can write it as z = ln(x2 + y2) − ln (x + y) ∂z 1 1 = 2 2. 2x − ∂x x + y x +y 2x2 + 2xy − x2 − y2 = (x2 + y2)(x + y) x2 + 2xy − y2 = 2 (x + y2)(x + y) Similarly, (or by symmetry) ∂z y2 + 2xy − x2 = ∂y (x2 + y2)(x + y) 36 © Copyright Virtual University of Pakistan 6-Geometry of continuous functions VU Example z = x 4 sin( xy 3 ) ∂z ∂ 4 = ⎡⎣ x sin( xy 3 ) ⎤⎦ ∂x ∂x ∂ ∂ = x 4 ⎡⎣sin( xy 3 ) ⎤⎦ + sin( xy 3 ) ( x 4 ) ∂x ∂x = x cos( xy ) y + sin( xy )4 x 4 3 3 3 3 ∂z = x 4 y 3 cos( xy 3 ) + sin( xy 3 ) ∂x ∂z ∂ 4 = ⎡⎣ x sin( xy 3 ) ⎤⎦ ∂y ∂y ∂ ∂ = x 4 ⎡⎣sin( xy 3 ) ⎤⎦ + sin( xy 3 ) ( x 4 ) ∂y ∂y = x 4 cos( xy 3 ).3 xy 2 + sin( xy 3 ).0 = 3 x5 y 2 cos( xy 3 ) Example z = cos(x5 y4) ∂z ∂ = − sin( x 5 y 4 ) ( x 5 y 4 ) ∂x ∂x = −5 x y sin( x5 y 4 ) 4 4 ∂z ∂ = − sin( x 5 y 4 ) ( x5 y 4 ) ∂y ∂y = −4 x y sin( x5 y 4 ) 5 3 Example w = x2 +3y2+4z2-xyz ∂w = 2x – yz ∂x ∂w = 6y - xz dy dw = 8z - xy dz 37 © Copyright Virtual University of Pakistan  7-Geometric meaning of partial derivative VU Lecture No -7 Geometric meaning of partial derivative Geometric meaning of partial derivative z = f(x, y) Partial derivative of f with respect of x is denoted by ∂z ∂f or fx or ∂x ∂x Partial derivative of f with respect of y is denoted by ∂z ∂f or fy or , ∂y ∂y Partial Derivatives Let z = f(x, y) be a function of two variable defined on a certain domain D. For a given change ∆x in x, keeping y as it is, the change ∆z in z, is given by ∆z = f (x + ∆x, y) – f (x, y) If the ratio ∆z f(x + ∆ x, y) − f(x, y) ∆x = ∆x approaches to a finite limit as ∆x →0, then this limit is called Partial derivative of f with respect of x. Similarly for a given change ∆y in y, keeping x as it is, the change ∆z in z, is given by ∆z = f (x , y + ∆y) – f (x, y) If the ratio ∆z f(x , y+ ∆y) − f(x, y) = ∆y ∆y approaches to a finite limit as ∆y →0, then this limit is called Partial derivative of f with respect of y. Geometric Meaning of Partial Derivatives Supposez = f(x, y) is a function of two variables.The graph of f is a surface. Let P be a point on the graph with coordinates (xo, yo, f(xo, yo)). If a point starting fromP, changes its position on the surface such that y remains constant, then the locus of this point is the curve of intersection of z= f(x, y) and y = constant.On this curve, ∂z is derivative ofz = f(x, y) with respect tox withy constant. ∂x ∂z Thus = slope of the tangent to this curve atP ∂x 38 © Copyright Virtual University of Pakistan 7-Geometric meaning of partial derivative VU ∂z Similarly, is the gradient of the ∂y tangent at P to the curve of intersection of z = f(x , y) and x = constant. As shown in the figure below (left) Also together these tangent lines are shown in figure below (right). Partial Derivatives of Higher Orders The partial derivatives f x and f y of a function f of two variables x and y, being functions of x and y, may possess derivati ves. In such cases, the second order partial derivatives are defined as below. ∂ ⎛∂f⎞ ∂ 2f ∂ ⎜ ⎟ = = (f ) = (fx )x = fxx = fx2 ∂ x⎝ ∂ x⎠ ∂ x2 ∂x x ∂ ⎛∂f⎞ ∂ 2f ∂ ⎜ = = (f ) = (fx )y = fxy ∂ y⎝ ∂ x⎠ ∂ y∂ x ∂y x ∂ ⎛∂f⎞ ∂ 2f ∂ ⎜ = = (f ) = (f y ) = f yx ∂ x⎝ ∂ y⎠ ∂ x∂ y ∂ x y x ∂ ⎛∂f⎞ ∂ 2f ∂ ⎜ ⎟ = 2 = (f ) = (f y)y = f yy = f y2 ∂y ⎝ ∂ y⎠ ∂y ∂y y Thus, there are four second order partial derivatives for a function z = f(x , y). The partial derivatives f xy and f yx are called mixed second partials and are not equal in general. Partial derivatives of order more than two can be defined in a similar manner. Example ⎛x ⎞ ∂ zz = arc1 sin ⎜⎜1 ⎟⎟. 1 =. ⎝ y= ⎠ ∂x x2 y y2 − x 2 1− 2 y ∂z 1 −x −x =. 2 = ∂y x2 y y y2 − x 2 1− 2 y ∂ 2z ∂ ⎛ ∂ z ⎞ − 1 2 2 − 3/2. −y = ⎜ = (y − x ) 2y = ∂ y∂ x ∂y⎝∂x ⎠ 2 (y − x2)3/2 2 39 © Copyright Virtual University of Pakistan 7-Geometric meaning of partial derivative VU ∂ 2z ∂ ⎛ ∂z⎞ = ⎜ ⎟ ∂x∂y ∂x⎝∂y ⎠ −1 x⎡ x ⎤ = − ⎢ 2 2 3/2⎥ y y2 − x 2 y ⎣(y − x ) ⎦ 2 2 2 − y +x − x −y = 2 2 3/2 = y(y − x ) (y − x2)3/2 2 Hence ∂ 2z ∂ 2z = ∂x∂y ∂y∂x Example f ( x, y ) = x cos y + ye x ∂f = cos y + ye x ∂x ∂2 f ∂ ⎛ ∂f ⎞ = ⎜ = − sin y + e x ∂y∂x ∂y ⎝ ∂x ⎠ ∂2 f ∂ ⎛ ∂f ⎞ = ⎜ = ye x ∂x 2 ∂x ⎝ ∂x ⎠ f(x, y) = x cosy + y e x ∂f = − x siny + ex ∂y ∂2 f = − siny + ex ∂x∂ y ∂2 f ∂ ⎛ ∂f ⎞ 2 = ⎜ = -x cos y ∂y ∂y ⎝ ∂y⎠ Laplace’s Equation For a function w = f(x,y,z) The equation ∂ 2f ∂ 2f ∂ 2f + =0 ∂ x2 ∂ y2 ∂ z2 is called Laplace’s equation. 40 © Copyright Virtual University of Pakistan 7-Geometric meaning of partial derivative VU Example f (x , y) = ex sin y + ey cos x, ∂f = ex sin y − ey sin x ∂x ∂ 2f x y 2 = e sin y − e cos x ∂x ∂f x y = e cosy + e cosx ∂y2 ∂f x y 2 = -e siny +e cosx ∂y Adding both partial second order derivative, we have ∂ 2f ∂ 2f x + = e sin y − eycos x ∂ x2 ∂ y2 x y − e sin y + e cos x =0 Euler’s theorem The mixed derivative theorem If f(x,y) and its partial derivatives fx, fy, fxy and fyx are defined throughout an open region containing a point (a, b) and are all continuous at (a, b), then fxy(a , b) = fyx (a , b) Advantage of Euler’s theorem ey w = xy + y2 +1 The symbol ∂2w/ ∂x ∂y tell us to differentiate first with respect to y and then with respect to x. However, if we postpone the differentiation with respect to y and differentiate first with respect to x, we get the answer more quickly. ∂w ∂ 2w =1 =y ∂y∂x ∂x Overview of lecture# 7 Chapter # 16 Partial derivatives Page # 790 Article # 16.3 41 © Copyright Virtual University of Pakistan 8-Euler theorem chain rule VU Lecture No- 8 More About Euler Theorem Chain Rule In general, the order of differentiation in an nth order partial derivatice can be change without affecting the final result whenever the function and all its partial derivatives of order ≤ n are continuous. For example, if f and its partial order derivatives of the firs t, second, and third ord ers are continuous on an open set, then at each point o f the set, f xyy = f yxy = f yyx or in ano ther notation. ∂3 f ∂3 f ∂3 f = = ∂y 2∂x ∂y∂x∂y ∂x∂y 2 Order of differentiation For a function f (x,y) = y2x4ex + 2 5 ∂3 f 2 ∂y ∂x If we are interested to find , that is, differentiating in the order firstly w.r.t. x and then w.r.t. y, calculation will involve many steps making the job difficult. But if we differentiate this function with respect to y, firstly and then with respect to x secondly then the value of this fifth order derivative can be calculated in a few steps. ∂5 f = 0 ∂ x 2∂ y 3 EXAMPLE x+ y f ( x, y ) = x− y ∂ ∂ ( x − y) ( x + y) − ( x + y) ( x − y) f x ( x, y ) = ∂x ∂x ( x − y) 2 ( x − y )(1) − ( x + y )(1) = ( x − y)2 − 2y = ( x − y )2 ∂ ∂ ( x − y) ( x + y) − ( x + y) ( x − y) f y ( x, y ) = ∂y ∂y ( x − y) 2 ( x − y )(1) − ( x + y)(−1) = ( x − y )2 2x = ( x − y)2 42 © Copyright Virtual University of Pakistan 8-Euler theorem chain rule VU EXAMPLE f ( x, y ) = x 3e − y + y 3 sec x 1 f x ( x, y ) = 3 x 2e − y + y 3 sec x tan x 2 x f y ( x, y ) = − x 3e− y + 3 y 2 sec x EXAMPLE f ( x, y ) = x 2 ye xy f x ( x, y) = 2 xye xy + x 2 y 2e xy = xye xy ( 2 + xy ) f x (1,1) = (1)(1)e(1)(1) [ 2 + (1)(1)] = 3e f ( x, y ) = x 2 ye xy f y ( x, y) = x 2 e xy + x 3 ye xy = x 2e xy (1 + xy) f y (1,1) = (1)(1)e (1)(1) [1 + (1)(1)] = 2e Example f (x, y) = x2 cos( xy) f x (x, y) = 2x cos( xy) − x2 y sin( xy) f x ( 1 ,π ) = 2( 1 ) cos(π ) − ( 1 )2 (π ) sin(π ) 2 2 2 2 2 = −π 4 f y ( x, y ) = − x3 sin( xy ) f y ( 12 , π ) = −( 12 )3 sin( π 2 ) = − 18 43 © Copyright Virtual University of Pakistan 8-Euler theorem chain rule VU EXAMPLE w =∂w( 4 x − 3 y + 2 z ) 5 = 20( 4 x − 3 y + 2 z ) 4 ∂x 2 ∂w =− 24 ( 4 x − 3 y + 2z ) 3 ∂y∂x ∂ 3w =−1440(4x−3y+2z)2 ∂z∂y∂x ∂4 w = −576 ( 4 x − 3 y + 2 z ) ∂z 2∂y∂x Chain Rule in function of One variable Given that w= f(x) and x = g(t), we find dw as follows: dt From w = f(x), we get dw dx From x = g(t), we get dx dt Then dw dw dx = dt dx dt Example w = x + 4, x = Sint By Substitution w = Sint + 4 dw = Cost dt dw w=x+4 ⇒ =1 dx dx x = Sint ⇒ = Cost dt By Chain Rule dw dw dx = × = 1. × Cost = Cost dt dt dt 44 © Copyright Virtual University of Pakistan 8-Euler theorem chain rule VU Chain rule in function of one variable y is a function of u, u is a function of v v is a function of w, w is a function of z z is a function of x. Ultimately y is function of x dy so we can ta lk about dx and by cha in rule it is given by dy dy du dv dw dz = dx du dv dw dz dx w = f(x,y), x = g(t), y = f(t) Dependent variable w = f(x,y) ∂w ∂w ∂y ∂x Intermediate variables x y dx dy dt dt dw ∂w dx ∂w dy = + dt ∂x dt ∂y dt t Independent variables EXAMPLE BY SUBSTITUTION w = xy x = cost, y = sint w = cost sint 1 = 2 sint cot 2 1 = sin2t 2 dw 1 = cos2t.2 dt 2 = cos 2t 45 © Copyright Virtual University of Pakistan  8-Euler theorem chain rule VU EXAMPLE w = xy, x = cos t , and y = sin t ∂w ∂w =x =y ∂y ∂x dx dy = − sint, dt = cost, dt dw ∂w dx ∂w dy = + dt ∂x dt ∂y dt = (sin t )( − sin t ) + (cos t )(cos t ) 2 = - sint + cos2 t = cos 2t EXAMPLE z = 3x2 y3 x = t4 , y = t2 ∂z ∂z = 6xy3 , = 9 x2 y2 ∂x ∂y dx dy = 4t3 , = 2t dt dt dz ∂z dx ∂z dy = + dt ∂x dt ∂y dt = (6xy3) (4t3) + 9x2y2 (2t) = 6 (t4) (t 6) (4t3) + 9 (t8) (t4) (2t) = 24 t13 + 18t13 = 42t13 EXAMPLE z = 1 + x- 2xy4 x = ln y=t ∂z 1 = ∂x 2 1 + x- 2xy4 ∂z 1 3 - 4xy3 =.(- 8xy) = ∂y 2 1 + x- 2xy4 1 + x- 2xy4 dx 1 d = , = dt t dt 46 © Copyright Virtual University of Pakistan 8-Euler theorem chain rule VU dz ∂z dx ∂z dy = +. dt ∂x dt ∂y dt 1 – 2y4 1 4xy3 = -.1 2 1 + x - 2xy t t 4 1 + x - 2xy4 1 ⎡ 1– ⎤ = ⎢ 4 4 ⎣ 2t - 4xy3 ⎥ 1 + x - 2xy ⎦ 4 1 ⎡1-2t ⎤ = ⎢ - 4 (lnt) t3 ⎥ 4 1 + lint - 2 (lnt) t ⎣ 2t ⎦ 1 ⎡1 3 3 ⎤ = ⎢ - t - 4t lnt⎥ 4 1 + lnt - 2t lnt ⎣2t ⎦ EXAMPLE z = ln (2x2 + y) 2/3 x = t, y = t ∂z 1 4 = 2. 4x = 2 ∂x 2x + y 2x + y ∂z 1 dx 1 1 dy 2 -1/3 = 2 , = , = t ∂y 2x + y dt 2 t dt 3 w = f(x,y,z), x = g(t) ,y = f(t), z = h(t) ∂w w = f(x,y,z) Dependent variable ∂w ∂x ∂w ∂z ∂y y z dy dx dz dt dt dt Independent variables t Overview of Lecture#8 Chapter # 16 Topic # 16.4 Page # 799 Book Calculus By Haward Anton 47 © Copyright Virtual University of Pakistan 8-Euler theorem chain rule VU Lecture No - 9 Examples First of all we revise the example which we did in our 8th lecture. Consider w = f(x,y,z) Where x = g(t), y = f(t), z = h(t) Then dw ∂w dx ∂w dy ∂w dz = + + dt ∂x dt ∂y dt ∂z dt Example: w = x2 + y + z + 4 x = e t, y = cost, z= t+4 ∂w ∂w ∂w = 2x, = 1, =1 ∂x ∂y ∂z dx dy dz = et , = −Sint, =1 dt dt dt dw ∂w dx ∂w dy ∂w dz = + +. dt ∂x dt ∂y dt ∂z dt t = (2x) (e ) + (1). (− Sint) + (1) (1) = 2 (et ) (et ) − Sint + 1 = 2 e2t − Sint + 1 Consider w = f(x), where x = g(r, s). Now it is clear from the figure that “x” is intermediate variable and we can write. ∂w dw ∂x and ∂w dw ∂x Dependent variable = = ∂r dx ∂r ∂s dx ∂s w = f(x) Example: dw 2 w = Sin x + x , x = 3r + 4s dx dw = Cosx + 2x dx x Intermediate variables ∂x ∂x =3 =4 ∂r ∂s ∂x ∂x ∂w dw ∂x ∂r =. ∂s ∂r dx ∂r = (Cosx + 2x). 3 = 3 Cos (3r+4s) + 6 (3r + 4s) s = 3 Cos (3r + 4s) + 18r + 24s ∂w dw ∂x =. ∂s dx ∂s = (Cosx + 2x). 4 = 4 Cosx + 8x = 4 Cos (3r + 4s) + 8 (3r + 4s) = 4 Cos (3r + 4s) + 24r + 32s 48 © Copyright Virtual University of Pakistan 9-Examples VU Consider the function w = f(x,y), Where x = g(r, s), y = h(r, s) w = f(x,y) Dependent variable ∂w ∂w ∂x ∂y x y Intermediate variables ∂x ∂r ∂x ∂y ∂y ∂s ∂r ∂s r s r r ∂w ∂w ∂x ∂w ∂y = + ∂r ∂x ∂r ∂y ∂r Similarly if you differentiate the function “w” with respect to “s” we will get And we have ∂w ∂w ∂x ∂w ∂y = + ∂s ∂x ∂s ∂y ∂s 49 © Copyright Virtual University of Pakistan  9-Examples VU Consider the function w = f(x,y,z), Where x = g(r, s), y = h(r,s), z = k(r, s) w = f(x,y,z) Dependent variable x y z intermediate variables ∂x ∂y ∂y ∂x ∂z ∂z ∂r ∂r ∂s ∂s ∂r ∂s r p s r r s Independent variables Thus we have ∂w ∂w ∂x ∂w ∂y ∂w ∂z = + + ∂r ∂x ∂r ∂y ∂r ∂z ∂r Similarly if we differentiate with respect to “s” then we have, ∂w ∂w ∂x ∂w ∂y ∂w ∂z = + + ∂s ∂x ∂s ∂y ∂s ∂z ∂s Example: r Consider the function w = x + 2 y + z 2 , x = , y = r 2 + ln s, z = 2r First we will calculate s ∂w ∂w =1 = 2 ∂w = 2 z ∂x = 1 ∂y = 2r ∂z = 2 ∂x ∂y ∂z ∂r s ∂r ∂r ∂w ∂w ∂x ∂w ∂y ∂w ∂z Now as we know that = + + By putting the values from above we get ∂r ∂x ∂r ∂y ∂r ∂z ∂r δw ⎛1⎞ = (1) ⎜ ⎟ + (2)(2r ) + (2 z )(2) δr ⎝s⎠ 1 1 = + 4r + (4r )(2) = + 12r s s Now ∂x r ∂y 1 ∂z =− 2 = =0 ∂s s ∂s s ∂s So we can calculate ∂w ∂w ∂x ∂w ∂y ∂w ∂z = + + ∂s ∂x ∂r ∂y ∂r ∂z ∂r ⎛ r ⎞ ⎛1⎞ = (1) ⎜ − 2 ⎟ + (2) ⎜ ⎟ + (2 z )(0) ⎝ s ⎠ ⎝s⎠ 2 r = − 2 s s 50 © Copyright Virtual University of Pakistan 9-Examples VU Remembering the different Forms of the chain rule: The best thing to do is to draw appropriate tree diagram by placing the dependent variable on top, the intermediate variables in the middle, and the selected independent variable at the bottom. To find the derivative of dependent variable with respect to the selected independent variable, start at the dependent variable and read down each branch of the tree to the independent variable, calculating and multiplying the derivatives along the branch. Then add the products you found for the different branches. The Chain Rule for Functions of Many Variables S up p o se ω = f ( x, y, …., υ ) is a d iffe re nt iab le fu nc t io n o f t he var iab les x, y, ….., υ (a fin ite set) a nd t he x, y, … , υ are d iffe re nt iab le fu n ct io ns o f p , q , , t (ano t her fin it e set ). T he n ω is a d iffe re nt iab le fu nc t io n o f t he var iab les p t hro u g h t a nd t he p art ia l d er iv at iv es o f ω w it h resp ect to t he se var iab le s are g iv e n b y eq ua t io ns o f t he fo r m ∂ω ∂ω ∂x ∂ω ∂y ∂ω ∂υ = + + …… +. ∂p ∂x ∂p ∂y ∂p ∂υ ∂p The other equations are obtained by replacing p by q, …, t, one at a time. One way to remember last equation is to think of the right- hand side as the dot product of two vectors with components. ⎛∂ω ∂ω ∂ω⎞ ⎛∂x ∂y ∂υ⎞ ⎜ , …… ⎟ and ⎜ , …… ⎟ ⎝ ∂x ∂y ∂υ ⎠ ⎝∂p ∂p ∂p ⎠ Derivatives of ω with Derivatives of the intermedaite respect to the variables with respect to the intermedaite variables selected independent variable Example: w = ln(e r + e s + et + eu ) Taking “ln” of both sides of the given equation we get e w = e r + e s + et + eu Now Taking partial derivative with respect to “r, s , u , and t” we get u −w , e wu = e ⇒ wu = e w u e w wr = e r ⇒ wr = e r − w , e w ws = e s ⇒ ws = e s − w and e w wt = et ⇒ wt = et − w 51 © Copyright Virtual University of Pakistan 9-Examples VU Now since we have wr = e r − w Now Differentiate it partially w.r.t. “s” wrs = er − w (− ws ) (Here we use the value of ws ) = −e r − w e s − w wrs = −e r + s − 2 w Now differentiate it partially w.r.t. “t” and using the value of wt we get, wrst = −e r + s − 2 w (−2 wt ) = 2e r + s − 2 w e t − w wrst = 2e r + s +t −3 w Now differentiate it partially w.r.t. “u” we get, wrstu = 2e r + s −3w (−3wu ) and by putting the value of w , we u get, wrstu = −6e r + s +t −3 w (eu − w ) wrstu = −6e r + s +t + u − 4 w 52 © Copyright Virtual University of Pakistan 11-The triple scalar or Box product VU Lecture No -10 Introduction to vectors Some of things we measure are determined by their magnitude. But some times we need magnitude as well as direction to describe the quantities. For Example, To describe a force, We need direction in which that force is acting (Direction) as well as how large it is (Magnitude). Other Example is the body’s Velocity; we have to know where the body is headed as well as how fast it is. Quantities that have direction as well as magnitude are usually represented by arrows that point the direction of the action and whose lengths give magnitude of the action in term of a suitably chosen unit. A vector in the plane is a directed line segment. B v A v = AB Vectors are usually described by the single bold face roman letters or letter with an arrow. The vector defined by the directed line segment from point A to point B is written as AB. Magnitude or Length Of a Vector : Magnitude of the vector v is denoted by v = AB is the length of the line segment AB Unit vector Any Vector whose Magnitude or length is 1 is a unit vector. Unit vector in the direction of vector v is denoted by v. and is given by v v= v Addition Of Vectors B b r A O a This diagram shows three vectors, in two vectors one vector OA is connected with tail of vector AB. The tail of third vector OB is connected with the tail of OA and head is connected with the head of vector AB.This third vector is called Resultant vector. 53 © Copyright Virtual University of Pakistan 11-The triple scalar or Box product VU The resultant vector can be written as r=a+b Similarly r=a+b+c+d+e+f e D E d f C F c r b a A B O Equal vectors Two vectors are equal or same vectors if they have same magnitude and direction. a a Opposite vectors Two vectors are opposite vectors if they have same magnitude and opposite direction. a -a Parallel vectors Two vector are parallel if one vector is scalar multiple of the other. b = λa where λ is a non zero scalar. 54 © Copyright Virtual University of Pakistan  11-The triple scalar or Box product VU r = x i+ y j + z k Addition and subtraction of two vectors in rectangular component: Let a = a1i + a2 j + a3k and b = b1i + b2 j + b3k a + b = (a1i + a2j + a3k) + (b1i + b2 j + b3k) = (a1 + b1 )i + (a2 + b2 )j + (a3 + b3)k a - b = (a1i + a2j + a3k) - (b1i + b2j + b3k) = (a1 - b1 )i + (a2 - b2 )j + (a3 - b3)k I