Consider the vector field F(x,y,z)=(2z+5y)i+(z+5x)j+(y+2x)k
a) Find a function ff such that F=∇f and f(0,0,0)=0
b)Suppose C is any curve from (0,0,0) to (1,1,1). Use part a) to compute the line integral ∫f.dr
2 Answers

f = ∫ (2z+5y) dx = 2xz + 5xy + g(y,z)
z + 5x = f_y = 5x + g_y(y,z)
. => . g_y(y,z) = z
g(y,z) = ∫ z dy = yz + h(z)
f = 2xz + 5xy + yz + h(z)
y + 2x = f_z = 2x + y + h'(z)
. => . h(z) = ∫ 0 dz = C
f = 2xz + 5xy + yz + C
Then use f(0,0,0) = 0 to find C=0, so
f = 2xz + 5xy + yz
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(b) Using the fundamental theorem of calculus, we have
∫ F.dr = f(1,1,1) – f(0,0,0) = 8

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