Chapter 16, Problem 13RCC

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) What is an oriented surface? Give an example of a non-orientable surface.

(b) Define the surface integral (or flux) of a vector field **F** over an oriented surface *S* with unit normal vector **n**.

(c) How do you evaluate such an integral if *S* is a parametric surface given by a vector function **r**(*u*, *v*)?

(d) What if *S* is given by an equation *z* = *g(x,* y)?

**(a)**

To determine

**To explain**: The oriented surface and example of a non-oriented surface.

Explanation

The oriented surface *S* is a surface which has a possibility choose a unit normal vector
*S* and the unit normal vector varies point to point continuously over the surface. The orientation of surface either positive or negative is decided by the orientation of choice of unit normal vectors over surface

**(b)**

To determine

**To define**: The surface integral of a vector field
*S* with unit normal vector

**(c)**

To determine

**To evaluate**: The surface integral of a vector field
*S* given by vector function

**(d)**

To determine

**To find**: The surface integral of vector field

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