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.

This quiz covers the concepts of tangent planes and normal lines, focusing on their equations at given points on surfaces. It also includes the calculation of the Jacobian matrix for multivariable functions, enhancing your understanding of partial derivatives and how they relate to function transformation.