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University of Dundee

Mathematics MA31007 Vector Calculus Quiz 4

Student ID Number: Please do not write your name on the paper. You have 12 minutes (plus additional time if applicable) to attempt all the questions in this quiz. There are questions on both sides of the paper and 10 marks in total. You may not refer to notes. Approved calculators are allowed. 1. (a) Fill in the blanks to complete the divergence theorem.

[3 Marks]

Let V be a closed bounded volume whose boundary is a piecewise (i) orientable surface S. Let v be a vector field that is continuous and has continuous first partial derivatives in some region containing V . Then ¨ v·n ˆ dS (ii) = S

where n ˆ is the (iii)

unit normal vector on S.

(b) Complete the following definition:

[1 Mark]

A volume is bounded if and only if. . .

(c) What is the volume element dV for cylindrical coordinates (r, φ, z)?

1

[1 Mark]

2. Evaluate the flux integral

¨

v · dS where v = xex + yey − 2zez and S is the part of the S

plane that is (Choose the surface normal that has a positive x-component.)

2

[5 Marks]...