a. A vector field exists within the region of space, R. It is F = ( x, y, z ). W

Question

a. A vector field exists within the region of space, R. It is F = ( x, y, z ). We wish to travel along the line in space defined by (x-2)/3 = (y-1)/4 = z/7 from x = 0 to x = 7. Calculate the line integral along that path. b. Calculate the line integral along three paths: 1. On the x-y plane from the starting point to ending point for z. 2. On the x-z plane from the starting point to ending point for y. 3. On the y-z plane from the starting point to ending point for x, (0, 7). You may do them in any order

Explanation / Answer

The vector field F is conservative. The potential function is f(x,y,z) = (x^2 + y^2 + z^2)/2 To be sure it is the potential function, just check that grad(f) = F. Since the vector field F is conservative, any line integral is independent of path. In this case, note that x=0 implies that y = 4(-2/3) + 1 = -5/3 z = 7(-2/3) = -14/3 Similarly, x=7 implies that y = 4(5/3) + 1 = 23/3 z = 7(5/3) = 35/3 With the starting point P(0, -5/3, -14/3) and the ending point Q(7, 23/3, 35/3), the value of the line integral is just f(Q) - f(P) = (7^2 + (23/3)^2 + (35/3)^2)/2 - (0^2 + (-5/3)^2 + (-14/3)^2)/2 Just pull out your calculator to see that the value is f(Q) - f(P) = 329/3.

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