Chapter 1. calc_tutorial_16_3_022

1.1 Problem Statement

{2,3}

{4,6,8}

{3,5,7}

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

1.2 Step 1

Question Sequence

Question 1.1

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 = <$a/x, −1/y> is conservative.

(∂F₁ / ∂y) = 1Wh3cvJ2xF4=

(∂F₂ / ∂x) = 1Wh3cvJ2xF4=

Thus F = <$a/x, −1/y> bfdtn5PFpMarVC8x2zfiEQ== conservative.

Correct.

Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Since F = <$a/x, −1/y> is conservative, it is also path independent. Thus any path from (1, 1) to ($b, $c) 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 $a/x = F₁ = − (∂V / ∂x).

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

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

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

(∂V / ∂y) = 1Wh3cvJ2xF4= + C_y(y)

Correct.

Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

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

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

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

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

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

Correct.

Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

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 ($b, $c) along any path in the first quadrant. Let c be any such path.

= V($b, $c) - V(1, 1) =

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

Correct.

Incorrect.