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Vector Calculus Independent Study

Unit 5 Sample Test

1. [25 points] Graph the vector field F⃗ (x, y) = (y, x^{2}). Be sure to include multiple points in each quadrant of the graph, and to sketch a few flow lines (assume F⃗ is a velocity vector field).

2. [15 points] Why isn’t F⃗ (x, y, z) = (−y, −x, x) a conservative field?

3. [20 points] Find a scalar potential for the gradient field F⃗ (x, y, z) = (z^{3} + 2xy, x^{2} + 2xy, 3xz^{2}).

4. [20 points] Calculate the divergence and curl of the vector field F⃗ (x, y, z) = (4xy, −x^{2}, 4z).

5. [20 points] Verify that the path σ⃗(t) = (sin(t), cos(t), e^{t}) is a flow line of the velocity vector field F⃗ (x, y, z) = (y, −x, z).