Functions of several variables

Visualization:

the n-dimensional real space ${\mathbb{R}}^n$ is defined as the set:

$${\mathbb{R}}^n=\{(x_1,x_2,...x_n)|x_i\in \mathbb{R}\}$$

Definition

Let D be a subset of ${\mathbb{R}}^n$. A real valued function f on D is a rule that assigns a unique real number $f(x_1,x_2,...x_n)$ to each point $(x_1,x_2,...x_n)\in D$

The graph of a function on ${\mathbb{R}}^2$

Definition

Let $f:D\subseteq {\mathbb{R}}^n\to \mathbb{R}$ be a real valued function. The graph of f is defined as the set:

$$\{(x_1,...,x_n,f(x_1,...,x_n))\in {\mathbb{R}}^{n+1}|(x_1,...x_n)\in D\}$$

• The graph of a function of n variables is a subset of (n + 1) dimensional real space ${\mathbb{R}}^{n+1}$
• The graph of a function of 2 variables is a surface in ${\mathbb{R}}^3$

Level curves

Definition

Let $f:D\to \mathbb{R}$ be a function with domain $D\subseteq {\mathbb{R}}^2$, and let c be a real number. The level curve at level c is the set.

$$\{(x,y)\in {\mathbb{R}}^2|f(x,y)=c\}$$

• the level curve can also be called a contour line

Limits of functions on ${\mathbb{R}}^2$

• 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 $f:D\to \mathbb{R}$ be a function of n variables and let $c\in D$ The function f is continuous at c if:

$$\underset{x\to c}{lim}f(x)=f(c)$$

Practical approach:

1. f(c) exists (c lies in the domain of f)
2. $\underset{x\to c}{lim}f(x)$ exists (f has a limit as x approaches c)
3. (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 $\underset{x\to c}{lim}f(x)$ 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:

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1. If a path limit along a continuous path ending at c does not exist, then $\underset{x\to c}{lim}f(x)$ does not exist.
2. If two paths limits along continuous paths ending at c do exist, but they are not the same then $\underset{x\to c}{lim}f(x)$ does not exist.

Polar coordinates:

• Every point (x, y) is described by two polar coordinates r and $\theta$
• The number r is called the radius, and is defined as the distance to (0, 0)
• the number $\theta$ 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:

$$x=r\cos \theta ,y=r\sin \theta$$

• From Cartesian coordinates to polar coordinates:

$$r=\sqrt{x^2+y^2},\theta =\arctan x,y$$

Limits and polar coordinates:

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Let f be a function of two variables.

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

$$f(r\cos \theta ,r\sin \theta )=r^{\alpha }g(\theta )$$

with:

• $\alpha >0$
• $g(\theta )$ bounded function.

then:

$$\underset{(x,y)\to (0,0)}{lim}f(x,y)=0$$

• 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:

$$\frac{d}{dx}f(x_0,y_0)=\underset{h\to 0}{lim}\frac{f(x_0+h,y_0)-f(x_0,y_0)}{h}$$

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

$$\frac{d}{dy}f(x_0,y_0)=\underset{h\to 0}{lim}\frac{f(x_0,y_0+h)-f(x_0,y_0)}{h}$$

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:

• $f_{xx}(x,y)$
• $f_{xy}(x,y)$
• $f_{yx}(x,y)$
• $f_{yy}(x,y)$

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

$$f_{xy}(a,b)=f_{yx}(a,b)$$

Linearization

The tangent plane

• The tangent plane V at $p=(x_0,y_0,f(x_0,y_0))$ is spanned by the vectors $u=(1,0,f_x(x_0,y_0))$ and $v=(0,1,f_y(x_0,y_0))$

• A normal vector is $n=v\cdot u$

• $n=(-f_x(x_0,y_0),-f_y(x_0,y_0))$

• A normal equation for V is $n\cdot (x-p)$ or

$$z-f(x_0,y_0)=f_x(x_0,y_0)\cdot (x-x_0)+f_y(x_0,y_0)\cdot (y-y_0)$$

Definition

The tangent plane is the graph of the linear function $L:{\mathbb{R}}^2\to \mathbb{R}$ defined by

$$L(x,y)=f(x_0,y_0)+f_x(x_0,y_0)\cdot (x-x_0)+f_y(x_0,y_0)\cdot (y-y_0)$$

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