Please solve all the questions below Let F be a vector field and g be a function

Question

Please solve all the questions below

Let F be a vector field and g be a function. Show that curl(g F) = g curl F + g Times F. For F(x, y, z) = (x, y2, z3), find div F, curl F. Show that the integral (2x ey dx + x2 ey dy) is independent on the path and find its value. Let C be the boundary of the domain enclosed between y = x2 and y = x. Assuming that C is oriented counterclockwise evaluate the integral (6xy + e-x2)dx (Additional and optional) Find a simple closed curve C with counterclockwise orientation that maximizes the value of (y3 dx + (3x - x3)dy and explain why.

Explanation / Answer

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