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Chapter 14: Q. 6 (page 1153)

Conservative Vector Fields: Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it.
$F (x, y) = <(1 + x y) e^{x y}, e^{x y}>$ .

Short Answer

Expert verified

$F (x, y)$ is not conservative.

Step by step solution

01

Step 1. Given Information.

The vector field is $F (x, y) = <(1 + x y) e^{x y}, e^{x y}>$ .

02

Step 2. Explanation.

A vector field $F (x, y) = <F_{1} (x, y), F_{2} (x, y)>$ is conservative if and only if $\frac{鈭� F_{1}}{鈭� y} = \frac{鈭� F_{2}}{鈭� x}$ , When $\frac{鈭� F_{1}}{鈭� y}$ and $\frac{鈭� F_{2}}{鈭� x}$ are both continuous on some open subset of $鈩�$ .

$\frac{鈭�}{鈭� y} ((1 + x y) e^{x y}) = 2 x e^{x y} + x^{2} y e^{x y}$

$\frac{鈭�}{鈭� x} (e^{x y}) = y e^{x y}$

so, $\frac{鈭� F_{1}}{鈭� y} 鈮� \frac{鈭� F_{2}}{鈭� x}$ ,

and partial derivatives are continuous on every open subset of $鈩�$ not containing the origin, $F (x, y)$ is not conservative.

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Most popular questions from this chapter

Compute the curl of the vector fields:

$F (x, y) = - 4 x^{2} y i + 4 x y^{2} j$ .

Use the same vector field as in Exercise 13, and compute the k-component of the curl of F(x, y).

Give an example of a field with positive divergence at (1, 0, 蟺).

$F (x, y, z) = i + j + k$ , where S is the lower half of the unit sphere, with n pointing outwards.

Integrate the given function over the accompanying surface in Exercises 27鈥�34.
$f (x, y, z) = x^{2} - y + 3 z$ , where Sis the portion of the plane with equation $2 x - 6 y + 3 z = 1$ whose preimage in the xz plane is the region bounded by the coordinate axes and the lines with equations z = 4 and x = z.

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