Find a function ff such that F=∇f and f(0,0,0)=0?

NetherCraft 0

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

    ——-

    (b) Using the fundamental theorem of calculus, we have

    ∫ F.dr = f(1,1,1) – f(0,0,0) = 8

  • K

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