AMath 231 ASSIGNMENT # 6: Surface Integrals, Spring 2014

Due on or before Thursday, June 19th at noon in the correct drop slot. This is the lastassignment before the midterm, and you might not get it back before then. Consider keepinga copy for study purposes. Solutions will be posted before the midterm.

1. Calculate the flux

F n dS and circulation

F dx for the given vector field F,

where is the triangular piece of the plane x + 2y + 3z = 6 with vertices on thecoordinate axes, and is the boundary of , oriented counterclockwise as seen fromthe point (0, 0, 4).

F = y2i+ j+ x2k

2. Suppose T (x, y, z) = x2 + y2 + z2 represents the temperature in a region of spacecontaining the origin of the coordinate system. Compute the heat flux across the unitsphere.

3. Verify the following property of the curl:

(FG) = ( G)F ( F)G+ (G )F (F )G

Note: It is sufficient to prove equality of the first component.

4. Starting with the expression (4.40) for in spherical coordinates in the course notes,derive the expression (4.49) for the divergence F of a vector field.Note: The question is tedious, but doing it builds character, I am told.

5. Elliptic coordinates (, ) in the plane are defined by the equation

(x, y) = (a cosh cos, a sinh sin),

where a is a positive constant.

i) Describe the two families of coordinate curves = constant and = constant,and illustrate them with a sketch.

ii) Calculate the coordinate basis vector fields e and e, and show that they aremutually orthogonal.