# Functions of several variables

## Visualization:

the n-dimensional real space

Definition

Let D be a subset of **real valued function** f on D is a rule that assigns a unique real number

## The graph of a function on

Definition

Let

- The graph of a function of n variables is a subset of (n + 1) dimensional real space
- The graph of a function of 2 variables is a surface in

### Level curves

Definition

Let **level curve** at level c is the set.

- the level curve can also be called a contour line

## Limits of functions on

- limit laws work the same way.
- The replacement rules are similar (instead of an interval surrounding c, use an open circular disc with center c)
- there is a sandwich info.
- definition and properties of continuity are automatically inferred.

### Continuity:

Definition

Let

Practical approach:

- f(c) exists (c lies in the domain of f)
exists (f has a limit as x approaches c) - (check that the limit equals the function value)

### Path limits

- Difference to functions of one variable: there are no one sided limits.

Definition

A path limit is the limit of f(x) where x approaches c along a continuous path ending in c.

info

The following statements are equivalent:

- (i) The limit
exists - (ii) All path limits of f(x) along continuous paths ending in c exist, and have the same value.

WARNING

In order to conclude i from ii you must show that path limits exist. and are the same along every possible path.

#### Showing that a limit does not exist:

info

- If a path limit along a continuous path ending at c does not exist, then
does not exist. - If two paths limits along continuous paths ending at c do exist, but they are not the same then
does not exist.

## Polar coordinates:

- Every point (x, y) is described by two polar coordinates r and
- The number r is called the radius, and is defined as the distance to (0, 0)
- the number
is called the polar angle, and is defined as the angle between the vector **r**and the positive x-axis.

### Coordinate transformation:

- From polar coordinates to Cartesian coordinates:

- From Cartesian coordinates to polar coordinates:

### Limits and polar coordinates:

info

Let f be a function of two variables.

If, after transformation to polar coordinates, f(x, y) can be written in the form:

with:

bounded function.

then:

- If the two requirements are not satisfied the limit most likely does not exist.

## Partial Derivatives

Definition

- The partial derivative of f with respect to x at (x0, y0) is:

- The partial derivative of f with respect to y at (x0, y0) is:

### How to calculate partial derivatives:

- The partial derivative with respect to x is the derivative of the function f(x, y) where y behaves like a constant.
- The partial derivative with respect to y is the derivative of the function f(x, y) where x behaves like a constant.

### Higher Order Partial derivatives of f:

### Mixed higher order partial derivatives:

Mixed Derivative info

If f(x,y) and it's partial derivatives exist and are continuous on an open environment of (a, b) then

## Linearization

### The tangent plane

The tangent plane V at

is spanned by the vectors and A normal vector is

A normal equation for V is

or

Definition

The tangent plane is the graph of the linear function

- this function is the linearization of f at (x0, y0)
- The function L is an approximation of f in the neighborhood of (x0, y0)