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For each of the following decide whether the vector field could be a gradient vector field. Be sure that you can justify your

Use the Fundamental Theorem of Line Integrals to calculate ∫CF⃗ ⋅dr⃗ exactly, if F⃗ =2x2/3i⃗ +ey/7j⃗ , and C is the quarter of the unit circle in the first quadrant, traced counterclockwise from (1,0) to (0,1).

If F⃗ is a path-independent vector field, with ∫(1,0)(0,0)F⃗ ⋅dr⃗ =5.8 and ∫(1,1)(1,0)F⃗ ⋅dr⃗ =2 and ∫(0,1)(0,0)F⃗ ⋅dr⃗ =4, find

The path C is a line segment of length 5 in the plane starting at (1,2). For f(x,y)=3x+4y, consider

Consider the vector field F(x,y,z)=(5z+3y)i+(z+3x)j+(y+5x)k.

The figure shows level curves of a function f(x,y).we