Although it is not defined on all of space R³, the field associated with the line integral below is defined on a region that is simply connected, and the component test can be used to show it is conservative. Find a potential function for the field and evaluate the integral. (4,4,4) 1 X y S + a + (1 - ) * - ** dx dy dz Z 2 (3,3,2) Find the potential function. f(x,y,z) =

Suppose f(x, y) = x² + 2xy+ y². Which field plot is Vf? Justify your answer.

For the vector field G = (yey + 3 cos(3x + y))i + (xeªy + cos(3x + y)) 3, find the line integral of G along the curve C from the origin along the x-axis to the point (2,0) and then counterclockwise around the circumference of the circle x² + y² = 4 to the point (2/√2,2/√√2). So G. dr =

Show that (vector)F = <ln(y) + y/x, ln(x) + x/y > is a gradient field. Then find a potential function f for (vector) F.

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Although it is not defined on all of space R³, the field associated with the line integral below is defined on a region that is simply connected, and the component test can be used to show it is conservative. Find a potential function for the field and evaluate the integral. (4,4,4) S (3,3,3) y 1 X y - + (-3)* - * dx dy dz z² Z Find the potential function. f(x, y, z)= Evaluate the line integral. (4,4,4) S (3,3,3) Y 1 dx + Z - X dy- dz=

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Find the line integral of the field F = х*z + 4xyz", х*у+ 4ху*z > Z over the straight-line path from (1,1,1,) to (2,2,2). Hint: Fis conservative! Find its potential function. 61 93 125 77

Consider the vector field 1 G(x, y, y, z) Cos y sin y 2√x' xsin y' 1+z² Show that G is conservative by finding all possible potential functions for G. (₂ X 2y tan y² : - 2e-² √VF). 2

Note: Show working / integration steps in details Given F = (2xz?e*4y +)i + (4z²e*+4y +)j + (2ze**++y – ) k y If F can be expressed as Grad V where V is a scalar field. Determine V.