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Consider the 3-dimensional vector field F defined by F(x,y,z)=(2xyz,x²z+2yz²,x²y+2y²z+e^z).

1.write down the Jacobian matrix jf(x,y,z).

2.determine divF (x,y,z).

3.determine curl F (x,y,z).

4.does F have a potential function? Give reasons for your answer, referring to the relevant definitions and theorems in the study guide.

5.find a potential function of F .

Find the curl F given that F x,y,z = 3x2i+2zj-xk

An example in the xy -plane. A force F=x^2yi+xy^2j acts on a body as it

moves between (0, 0) and (1, 1). Determine the work done when the path is along the line y=x.

Select one:

A. 1

B. 14

C. 13

D. 12

Evaluate ∇2U, given that Ux,y,z=xy2z3

Select one:

A. y2z3+2xz3+6xy2z

B. x2y2z+2xz3-6xy2z

C. y2z3+2xyz3+3xy2z2

D. 2xz3+6xy2z

Compute the divergence of the vector field exy2i+x+2yj

Select one:

A. exy2+2

B. y2+2

C. 2exy

D. exy2+2x+y

verify stoke’s theorem when f = x ^ 2 i + y ^2 j + z ^ 2k ,s is the upper hemisphere

z =√( a ^2 − x ^2 − y ^2)

Determine whether 𝐹 is conservative vector field. If so, find a potential

function for it is

f(x,y,z)=x^yi+5xy^2j

Let E:=Exi^+Eyj^+Ezk^ and H:=Hxi^+Hyj^+Hzk^ be two vectors assumed to have continuous partial derivatives (of second order at least) with respect to position and time. Suppose further that E and H satisfy the equations:∇⋅E=0,∇⋅H=0,∇×E=−1c∂H∂t,∇×H=1c∂E∂tprove that E and H satisfy the equation∇2Ei=1c2∂2Ei∂t2 and ∇2Hi=1c2∂2Hi∂t2Here, i=x,y or z.

Let r=xi^+yj^+zk^ and r=r .Show that:∇(lnr)=rr2. and∇×(rnr)=0.

Find the angles between the force F = 1200𝐢 + 800𝐣 − 1500𝐤 N and the x-, y-, and z-axes. Show your results on a sketch of the coordinate system.

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