Intro to Multivariable Calculus & 3D shapes

vector-valued functions

function that takes a single variable input and outputs a vector

multivariable function

function that takes a list of numbers as input and outputs one (or more) numbers

Function of two variables (x & y)

assigns the pair (x,y) to exactly one output

Domain

set of pairs of (x,y) of inputs for which the function is definedci

Circle Equation

(x-h)² + (y-k)² = R²

Ellipse Equation

(x-h)²/b² + (y-k)²/a² = 1

Hyperbola

(y-k)²/a² - (x-h)²/b² = 1

Parabola Equation

(y-k) = (x-h)²

What axis does z² - y² = R open to?

z-axis

What axis does y² - z² = 1 open up to?

y-axis

Graph of f(x,y)

x out; y horizontal in board; z up

Trace

replace independent variable with constant

Trace in x-direction

set y constant

Trace in y direction

set x constant

Level Curve

aka contour; shape obtained by setting z constant

Contour diagram

collection of multiple contours on the same graph

What is the 3D shape of x² + y² - z = 0?

paraboloid

What are a paraboloid’s traces?

parabolas

What are a paraboloid’s level curves?

circles

What do adding constants do to a paraboloid’s trace?

shift up/down

What do coefficients do to a paraboloid’s level curves?

change shape to ellipses

What parts of the equation hint that the shape is a paraboloid?

2 variables are squared, one is not

What is the 3D shape of x² + y² - z² = 0?

cone

What is the shape of a cone’s traces?

lines & hyperbolas

What is the shape of a cone’s level curves?

circles

What are the equation’s hints for a cone?

3 variables are squared & equal to zero

What is the 3D shape of -x² - y² + z² = -1?

hyperboloid of one sheet

What is the shape of a hyperboloid of one sheet’s traces?

lines & hyperbolas

What is the shape of a hyperboloid of one sheet’s level curves?

circles

What are the hints from equations for hyperboloid of one sheet?

odd amount of signs (aka 3+ & 1- or 3- & 1+)

What is the 3D shape of -x² - y² + z² =1?

hyperboloid of two sheets

What is the shape of a hyperboloid of two sheets’ traces?

hyperbolas

What is the shape of a hyperboloid' of two sheet’s level curves?

circles

What are the hints in an equation for a hyperbola of two sheets?

same signs (aka 2+ & 2-)

What is the 3D shape of x² + y² + z² = 1?

sphere

What is the shape of a sphere’s traces?

circlesl

What is the shape of a sphere’s level curves?

circles

What are the hints in equations of spheres?

3 variables are squared