Chapter 16, Problem 13RE

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Show that **F** is a conservative and use this fact to evaluate ∫_{C}**F** · *d***r** along the given curve.

**13. F**(*x*, *y*) = (4*x*^{3}*y*^{2} – 2*xy*^{3}) **i** + (2*x*^{4}*y* – 3*x*^{2}*y*^{2} + 4*y*^{3}) **j**, *C:* **r**(*t*) = (*t* + sin *πt*) **i** + (2*t* + cos *πt)* **j**, 0 ⩽ *t* ⩽ 1

To determine

**To show: F** is a conservative vector field and evaluate the value of

Explanation

**Given data:**

**Formula used:**

Consider the standard equation of curl **F** for

If the curl of a vector field **F** is zero, then **F** is said to be a conservative vector field.

The relation between the potential function *f* and vector field

Obtain the value of

Substitute

On simplification,

Since **F** is a conservative vector and the domain of **F** is

Thus, **F** is a conservative vector field.

Substitute

Compare the equation (4) and equation (1).

Integrate equation (5) with respect to *x*.

Apply partial differentiation with respect to *y* on both sides of equation (7)

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