Introduction to Engineering Mathematics

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Vector Calculus for Engineers

CME100, Fall 2004

Handout #4

Line Integrals, Surface Integrals, Green’s, Divergence, and Stokes’ Theorems

1. A semi-circular loop of wire of radius 1m and carrying current [pic] in the counterclockwise direction is subjected to a magnetic field in the z-direction such that [pic]. Determine the net force acting on the wire segment.

2. Find the center of mass of an object constructed from two uniform segments [pic] and [pic], where x ranges from 0 to 1.

3. Find the work done by the force [pic] around the circle: [pic], where [pic].

4. Find the flux of [pic] around the circle: [pic], where [pic].

5. Find the work done by [pic] from (0,0,1) to (1,1,1):

a) by parametrizing the curve [pic]

b) by parametrizing the curve [pic]

c) using the scalar potential

6. Show that [pic] is conservative and find the scalar potential for:

[pic]

7. Evaluate [pic] over the square defined by [pic]

a) by parametrizing the curve

b) using Green’s theorem

8. Verify the circulation form of Green’s theorem on the annular ring: [pic], if:

[pic]

9. Find the surface area of a hemisphere of radius R

10. Find the flux [pic] of [pic] across the surface of a hemisphere of radius R

11. Find the flux [pic] of [pic] through the surface of a cylinder [pic] bounded by [pic] and [pic] by inspection.

12. Calculate the flux [pic] of [pic] through the surface of the cube cut from the first octant by the planes [pic], [pic], and [pic]. Then verify your result using the divergence theorem.

13. Use Stokes’ theorem to evaluate the line integral of [pic] over the circle [pic]:

a) using parametrization

b) using the disk in the xy-plane

c) using the upper hemisphere of [pic]

d) using the surface of the paraboloid [pic]

14. Using Stokes’ theorem, evaluate the line integral [pic] if [pic] over the curve defined by the portion of the plane [pic] in the first octant, traversed counterclockwise.

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