Multivariable Calculus

Multivariable Calculus

0. There two fundamental topics in calculus: Derivatives, which study the rate of change of a function as you tweak its input.Integrals, which study how to add together infinitely many infinitesimal quantities that make up a function's output

1. Multivariable calculus is all about abstracting the ideas of differentiation and integration from the familiar single variable case to that of higher dimensions.

2. Deals with functions that handle multivariables - multivariable input and outputs. If there are multiple variables in the output its a vector.
3. In multivariable calculus, we have double integrals, triple integrals, line integrals, surface integrals

4. Visualization - can be visualized in three dimensions or 2D.  Example parametric surfaces, vector fields,

Functions
Functions whose output is a vector are called vector-valued functions

Representing points in 3D

1. Think of pairs of numbers or triplets as points in space

Vectors Calculus

Line integral - which can be used to find the work done by a force field in moving an object along a curve.

Surface integrals - which can be used to find the rate of fluid flow across a surface

Introduction to 3D Graphs

3D graphs are function of x and y i.e f (x,y). Example f(x,y) = cos(x) sin(y) or z = cos(x) sin(y)

visualization helps understand the relationship between inputs and outputs. Not all multivariate functions lend themselves to 3D graphs.. example function have a 3D I/O and 2/D O/P

Another way to visualize is through contour maps

Interpreting graphs with slices

Holding one variable constant helps understand the 3D object better

Contour Plot

1. Slice the 3D object to slice it parallel to xy plane. 

2. Contour plot with squished 

Parametric Curves

There are also a great many curves out there that we can’t even write down as a single equation in terms of only $x$ and $y$. 

So, to deal with some of these problems we introduce parametric equations. Instead of defining $y$ in terms of $x$ ($y=f\left(x\right)$) or $x$ in terms of $y$ ($x=h\left(y\right)$) we define both $x$ and $y$ in terms of a third variable called a parameter as follows  x= f(t) y = g(t)

Parametric Surfaces

Has a 2D I/O and 3D O/P.. examples Torus

Vector Field

• A vector field associates a vector with each point in space.
• Vector field and fluid flow go hand-in-hand together.
• You can think of a vector field as representing a multivariable function whose input and output spaces each have the same dimension.
• The length of arrows drawn in a vector field are usually not to scale, but the ratio of the length of one vector to another should be accurate. Sometimes vector length is communicated using color.

Transformations

Partial Derivatives

Vector Calculus

Line integrals - which can be used to find the work done by a force field in moving an object along a curve. 

Surface integrals - which can be used to find the rate of fluid flow across a surface

Vector Fields

Example - Ocean Currents

Example -  Force Field - associates a force vector at each point in a region

In general a vector field is a function whose domain is a set of points in R2 or R3, Range is a set of vectors in V2 or V3