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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 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 dimensional column vector of all the firstordered partial derivatives ( 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:
 We can think of as an operator full of partial derivative operations:
Directional derivative = tells us how a “nudge” in the direction of a particular vector changes the output of a multivariable function
 General equation in 2D:
 For ,

 Normalized formula for the directional gradient:
Laplacian = the divergence of the gradient of a MISO (multiinput, singleoutput) system
 General equation:
 Note: harmonic functions are defined as MISO systems that have for all input vectors
Multivariable Chain Rule
Used to determine the derivative of a multivariable function composed of singlevalued 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 singlevalued functions) with respect to $t$ (i.e., the input variable for our singlevalued 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