calc_tutorial_16_3_12

Question 1

Recall the cross-partial property of conservative vector fields.

F = <F₁, F₂, F_3> is conservative if and only if (∂F₁ / ∂y) = (∂F₂ / ∂x), (∂F₂ / ∂z) = (∂F₃ / ∂y), and (∂F₃ / ∂x) = (∂F₁ / ∂z).

Use this property to check if F = <cos z, 14y, −x sin z> is conservative.

(∂F₁ / ∂y) = (∂F₂ / ∂x) =

(∂F₂ / ∂z) = (∂F₃ / ∂y) =

(∂F₃ / ∂x) = --------sin z-sin z (∂F₁ / ∂z) = ---------sin zsin z

Thus F = <cos z, 14y, −x sin z> --------isis not conservative.

Question 2

We wish to find a potential function, V(x, y, z), such that F = ∇V. Thus cos z = F₁ = (∂V / ∂x).

Integrate (∂V / ∂x) = cos z with respect to x, holding y and z fixed.

= --------x sin zx cos zz cos z + C(y, z)

We need to determine C(y, z). To do so, differentiate this expression with respect to y.

(∂V / ∂x) = + C_y(y, z)

Question 3

Since F = ∇V, we have 14z = F₂ = (∂V / ∂y) = C_y(y, z).

Integrate C_y(y, z) = 14y with respect to y, holding z fixed.

= y² + D(z)

We now have V(x, y, z) = x cos z + y² + D(z), and need to determine D(z). To do so, differentiate this expression with respect to z.

(∂V / ∂z) = ---------x sin z-x cos zx sin zx cos z + D_z(z)

Question 4

Since −x sin z = F₃, use the condition (∂V / ∂z) = F₃ to solve for D_z(z).

D_z(z) =

Integrate D_z(z) with respect to z.

D(z) =

A.

B.

C.

D.

= + C

Thus the general potential function is V(x, y, z) = --------x cos z0x sin z + 7y² + C.