calc_tutorial_16_3_022

Problem Statement

{2,3}

{4,6,8}

{3,5,7}

Let F = <3/x, −1/y>. Calculate the work against F required to move an object from (1, 1) to (4, 5) along any path in the first quadrant.

Step 1

Question Sequence

Question 1

Recall the cross-partial property of conservative vector fields.

F = <F₁, F_2> is conservative if and only if (∂F₁ / ∂y) = (∂F₂ / ∂x). Use this property to check if F = <3/x, −1/y> is conservative.

(∂F₁ / ∂y) =

(∂F₂ / ∂x) =

Thus F = <3/x, −1/y> conservative.

Correct.

Incorrect.

Step 2

Question Sequence

Question 2

Since F = <3/x, −1/y> is conservative, it is also path independent. Thus any path from (1, 1) to (4, 5) will yield the same amount work. In order to evaluate the line integral representing the work, we need to find a potential function, V(x, y), such that F = −∇V (the minus sign is a convention from physics). Thus 3/x = F₁ = − (∂V / ∂x).

Integrate (∂V / ∂x) = −3/x with respect to x.

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

(∂V / ∂y) = + C_y(y)

Correct.

Incorrect.

Step 3

Question Sequence

Question 3

Since F = −∇V, we have −1/y = F₂ = − (∂V / ∂y) = −C_y(y).

Integrate C_y(y) = 1/y with respect to y.

Thus the general potential function is V(x, y) =

Correct.

Incorrect.

Step 4

Question Sequence

Question 4

Recall the fundamental theorem for conservative vector fields.

If F = −∇V and c is a path from P to Q, then .

Use the fundamental theorem for conservative vector fields to determine the amount of work against F required to move an object from (1, 1) to (4, 5) along any path in the first quadrant. Let c be any such path.

= V(4, 5) - V(1, 1) =

Correct.

Incorrect.