Functions of several variables

Function, Domain and Range

Domain

Is a solution of

Vector CalculusDefinition The Euclidean norm (or simply norm) of a vector

x = is defined as

Properties

The Scalar Product

Definition

DefinitionTwo vectors x and y are called orthogonal or perpendicularif x · y = 0, and we write x y in this case.

Examine whether the vectors x = (2, 1, 1) and y = (1, 1,−3) are orthogonal. We have x · y = 2 · 1+1 · 1+1 · (−3) = 2+1−3 = 0. This implies x y.

Definition Let x, y be vectors with y 6= 0. The projection of x ony, denoted by py (x), is defined by

The length of the projection is given by

Definition

Example Find the angle between the vectors x = (2, 3, 2) and y = (1, 2,−1).

Cross Product

The magnitude of x × y equals the area of that parallelogram, so

Moreover, x × y is orthogonal to both x and y.

Right-hand rule: Point the index finger in the direction of x and the middle finger in the direction of y. The thumb then points in the direction of x × y.

Example. Calculate x × y where x = (1,−2, 3) and y = (2, 1,−1).

Differential Calculus of Vector Fields

Stationary

Instationary

Let f1(t) = 2 cos t, f2(t) = 2 sin t, f3(t) = t. Write down the associatedvector field having f1, f2 and f3 as components.

Definition : Derivative of a vector field

Example

Solution

(0)

Vector Fields in Several Dimensions

Example

Definition (Directional Derivative)

Example

Solution

Theorem