Calculus: Tangent Planes and Jacobians

Tangent Planes and Normal Lines

• Find the equation of the tangent plane and normal line to a surface at a given point.
• The equation of the tangent plane to a surface is found by taking the gradient of the surface equation at that point and setting it equal to zero.
• For the surface $x^2 + y^2 - 4z = 5$ at point $(3,4,5)$:
  • Find the gradient of the surface equation: $\nabla(x^2 + y^2 - 4z) = (2x, 2y, -4)$.
  • Evaluate the gradient at the point $(3,4,5)$: $\nabla(3,4,5) = (6, 8, -4)$.
  • The equation of the tangent plane is: $6(x - 3) + 8(y - 4) - 4(z - 5) = 0$.
  • To find the equation of the normal line, use the gradient as the direction vector and the given point as the starting point.

Jacobian

• Calculate the Jacobian matrix for multivariable functions.
• The Jacobian matrix is a matrix of partial derivatives of a vector-valued function.
• For the function $u = x \sin y$, $v = y \cos x$:
  • The Jacobian matrix is:
    • $\begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{pmatrix} = \begin{pmatrix} \sin y & x \cos y \ -y \sin x & \cos x \end{pmatrix}$

Lagrange Multipliers

• Find maximum and minimum values of a function subject to a constraint.
• The method of Lagrange multipliers is a technique for finding the maxima and minima of a function subject to one or more constraints.
• To find the maximum value of $f(x, y) = 2x +y$ on the circle $x^2 + y^2 = 1$:
  • Construct the Lagrange function, $L(x, y, \lambda) = f(x,y) - \lambda g(x,y)$, where $g(x,y)$ is the constraint function.
  • $\nabla L = (0, 0, 0)$
  • The system of equations to solve is:
    • $2 - 2\lambda x = 0$
    • $1 - 2\lambda y = 0$
    • $x^2 + y^2 = 1$
  • From the first two equations, $\lambda = \frac{1}{x}$ and $\lambda = \frac{1}{2y}$, which implies $x = 2y$.
  • Substitute $x = 2y$ into the constraint equation to get $y^2 = \frac{1}{5}$.
  • Substitute $y = \pm \sqrt{\frac{1}{5}}$ into the constraint equation to find $x$.
  • Evaluate $f(x,y)$ at the critical points to find the maximum and minimum values.

Optimization problems in 3D Space

• Find the maximum or minimum value of a function in 3D space subject to a constraint.
• For example, find the highest pressure on the surface of the unit sphere $x^2 + y^2 + z^2 = 1$ given the pressure at any point $(x,y,z)$ is $P = 400xyz^2$.
  • Use the method of Lagrange multipliers, similar to the previous example.
  • Construct the Lagrange function and solve for the critical points.
  • Evaluate the pressure function at these critical points to find the maximum pressure.