Consider the following vector field. F(x, y, z) = (2e* sin(y), 8e) sin(z), 5e² sin(x) (a) Find the curl of the vector field. curl(F) = (5e³cos(x), – 5e²cos(x),16e'sin(z) — 2e *cos(y)) (b) Find the divergence of the vector field. div(F) = 2e* cos(y) + 8e³cos(z) + 5e³cos(x) ×

Let F and G be vector-valued functions such that F(t) = (cos(nt), e²t-1, t² – 1), Ġ(1) = (1, 1, – 1), Ğ'(1) = (2,3, 2), Ġ"(1) = (0,1,0) i. Find a vector equation of the tangent line to the graph of F at (-1, e,0). ii. Evaluate (F · G)'(1). iii. Evaluate (G x G')'(1). d'(1) = (2,3, 2), Ğ"(1) = (0,1,0) %3D

Consider the vector field. F(x, y, z) = (4e* sin(y), 5e sin(z), 5e² sin(x)) (a) Find the curl of the vector field. curl F = -2e*cos(y) x (b) Find the divergence of the vector field. div F = 4ex cos (y) + 5ey cos (z) + 5ez cos (x) x

Q3: ( (a) Prove that div(ū x w)= w•curl ū- ū• curl w functions. where u and w are vector (b)Find the derivative of g in the direction of a 2i+curl w at point (0,-1,1), where g = In- u +2 u = z(xy + z?, W = cos-i+xe"*l yz

3. Let f(x, y) = sin x + sin y. (NOTE: You may use software for any part of this problem.) (a) Plot a contour map of f. (b) Find the gradient Vf. (c) Plot the gradient vector field Vf. (d) Explain how the contour map and the gradient vector field are related. (e) Plot the flow lines of Vf. (f) Explain how the flow lines and the vector field are related. (g) Explain how the flow lines of Vf and the contour map are related.

Consider the following vector field. F(x, y, z) = (4e* sin(y), 9e sin(z), 8e? sin(x)) (a) Find the curl of the vector field. curl(F) = (b) Find the divergence of the vector field. div(F)

Find g(x, z) so that F(x,y,z) = (8x cos y sin? z + 2e2xz)î – (4x² sin y sin? z)j + (g(x, z) cos y + 2e2xz)k is a gradient vector field.

The figure shows the vector field F (x,y) = {2xy,x2 } and three curves that start at (1,2) and end at (3,2). (a) Explain why∫c F. dr has the same value for all the three curves. (b) What is this common value?

2. Let x' = [5, 1, 3] and y' = [−1, 3, 1]. (a) Find the inner product, y'x. (b) Find x - 2y. (c) Find the length of x, Lx. (d) Find the length of y, Ly. (e) Find the unit vector of x (i.e., the unit vector with the same direction as x). (f) Find the unit vector of y.

F = (x² - y)i + + (4z)j + (x²)k Find the curl (curl calculation) of the vector field?