Explain why or why not Determine whether the

QUESTION:

Explain why or why not Determine whether the following stalements are true and give an explanation or counterexample.

a. A paddle wheel with its axis in the direction \(\langle 0,1,-1\rangle\) would not spin when put in the vector field \(\mathbf{F}=\langle 1,1,2\rangle \times\langle x, y, z\rangle\).

b. Stokes' Theorem relates the flux of a vector field F across a surface to the values of F on the boundary of the surface.

c. A vector field of the form \(\mathbf{F}=\langle a+f(x), b+g(y), c+h(z)\rangle\). where a. b, and c are constants, has zero circulation on a closed curve.

d. If a vector field has zero circulation on all simple closed smooth curves C in a region D. then F is conservative on R.

Text Transcription:

Langle 0, 1, -1 rangle

F = langle 1, 1, 2 rangle x langle x, y, z rangle

F = langle a + f(x), b + g(y), c + h(z) rangle