Mathematics

Multivariable Calculus

[latexpage]

Visualizing multivariable functions

Parametric surface = surfaces in 3D space that are actually “maps” of somje 2D space

  • Start with a 2D input that somehow “moves” into 3D
  • e.g., nraveling a cylinder into a rectangular plane

Vector fields = every point in the input space is associated with a vector in the output

  • Used for visualizing functions that have the same number of dimensions in their input as their output
  • Note: often, output vectors are scaled down to get a much better (and cleaner) feel for the output corresponding with each input point
    • Importantly, this is kind of like lying, so we should always indicate when we are scaling down our vectors
  • Useful for “fluid flow” interpretations

Contour maps = allows us to visualize 3D graphs in a 2D setting (associate each input point with a color that indicates the magnitude of the output)

  • Contour lines = lines that are used to denote constant output values on a contour map
    • Can “squish” contour lines together along an xy-plane to get a 2D representation of some (but not all) of our outputs
    • Each line represents a constant output of our function
  • Usually, “warmer” colors are used to represent higher output values and “cooler” colors are used to represent lower output values

Operators

Gradient = a way of packing all the partial derivative information of a function into a vector

  • Mathematically defined as an arbitrary n-dimensional column vector of all the first-ordered partial derivatives (n is equal to the number of inputs an arbitrary function takes)
  • Note: for any arbitrary multivariable function, the gradient is always the direction of steepest ascent
  • General definition of the gradient of a multivariable function:

\nabla f(x_1 , ..., x_n) = \begin{bmatrix} \frac { \partial f}{ \partial x_{ 1 } } \\ \vdots \\ \frac { \partial f}{ \partial x_{ n } } \end{bmatrix}

  • We can think of \nabla as an operator full of partial derivative operations:

\nabla = \begin{bmatrix} \frac { \partial }{ \partial x_{ 1 } } \\ \vdots \\ \frac { \partial }{ \partial x_{ n } } \end{bmatrix}

Directional derivative = tells us how a “nudge” in the direction of a particular vector \overrightarrow{v} changes the output of a multivariable function

  • General equation in 2D:
    • For \overrightarrow{v} = \begin{bmatrix} a \\ b \end{bmatrix},

\nabla _{\overrightarrow{v}} f = \overrightarrow{v} \cdot \nabla f = a \frac{\partial f}{\partial x} + b \frac{\partial f}{\partial y}

    • Normalized formula for the directional gradient:

\nabla _{\overrightarrow{v}} f = \frac{ \overrightarrow{v} \cdot \nabla f}{\left\| \overrightarrow{v} \right\| }

Laplacian = the divergence of the gradient of a MISO (multi-input, single-output) system

  • General equation: \Delta f(\overrightarrow{x}) = \nabla \cdot \nabla f(\overrightarrow{x})
  • Note: harmonic functions are defined as MISO systems that have \Delta f(\overrightarrow{x}) = 0 for all input vectors \overrightarrow{x}

Multivariable Chain Rule

Used to determine the derivative of a multivariable function composed of single-valued functions (that have the same input variable) with respect to that variable

Prototypical example: let $x(t)$ and $y(t)$ be inputs to a 2D multivariable function $f(x(t),y(t))$

  • The multivariable chain rule says that in general, the derivative of $f(x(t),y(t))$ (i.e., the composition of multiple single-valued functions) with respect to $t$ (i.e., the input variable for our single-valued functions) will be the sum of the products between the partial derivative of our MVF with respect to one of its inputs and the derivative of that input with respect to $t$:

$\frac{df}{dt}(x(t),y(t)) = \frac{\partial f}{\partial x} \frac{dx}{dt}+ \frac{\partial f}{\partial y} \frac{dy}{dt}$

Sources

  • Khan Academy – Multivariable Calculus

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