University Mathematics Handbook (2015)

IX. Vector Analysis

Chapter 6. Vector Functions of Two Variables

a.  Vector

 (*)

is a vector function dependent of variables and and defined in domain .

b.  The graph of a two-variable vector function is a surface in a three-dimensional space, presented in a parametric form

vise versa, any surface in space can be presented in a vector form (*).

Example:

1.  Using spherical coordinates, the vector representation of sphere is

2.  Surface presented in a vector form:

c.  Normal to a Surface

1.  If be a point on surface , then vector is in the direction of the tangent line to curve and belongs to the tangent plane to the surface at .

Vector is in the direction of the tangent line to curve , which is on the tangent plane to the surface at the same point, .

Normal , which is perpendicular to the tangent plane, that is, to the plane of vectors and , is

The unit vector of the normal is

2.  Normal to a surface, given in its explicit form of is

3.  Normal to a surface presented in the form of is