Functions of several variables

Function, Domain and Range

Domain

Is a solution of

Vector Calculus DefinitionThe Euclidean norm (or simply norm) of a vector x = is defined as Properties The Scalar Product Definition

Two vectors x and y are called orthogonal or perpendicular if 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 on y, denoted by p y (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 f 1 (t) = 2 cos t, f 2 (t) = 2 sin t, f 3 (t) = t. Write down the associated vector field having f 1, f 2 and f 3 as components.

Definition: Derivative of a vector field

Example Solution (0)

Vector Fields in Several Dimensions Example

Definition (Directional Derivative) Example Solution Theorem