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Found in: Page 760

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

## $$f(x,y) = x{e^{xy}}$$

The gradient vector field of $$f$$ is $$\overline V f\left( {x,y} \right) = {e^{xy}}\left( {1 + xy} \right)i + \left( {{x^2}{e^{xy}}} \right)j$$.

See the step by step solution

## Step 1: Given Information.

It is given that $$f(x,y){\rm{ }} = {\rm{ }}x{e^{xy}}$$.

## Step 2: Find the gradient vector field.

$$f(x,y){\rm{ }} = {\rm{ }}x{e^{xy}}$$

$$\overline V f\left( {x,y} \right) = {f_x}\left( {x,y} \right)i + {f_y}\left( {x,y} \right)j$$

Applying the formula to the given function and using summation derivation rule, we get

\begin{aligned}\overline V f\left( {x,y} \right) &= {f_x}\left( {x,y} \right)i + {f_y}\left( {x,y} \right)j \\ &= \frac{\partial }{{\partial x}}\left( {x{e^{xy}}} \right)i + \frac{\partial }{{\partial y}}\left( {x{e^{xy}}} \right)j \\ &= \frac{\partial }{{\partial x}}\left( {\mathop{\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{x} }\limits_u \mathop {{{\mathop{\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{e} }\limits_v}^{xy}}}\limits_{} } \right)i + x\frac{\partial }{{\partial y}}\left( {{e^{xy}}} \right)j\\&= \left( {1 \cdot {e^{xy}} + x \cdot y{e^{xy}}} \right)i + x\frac{\partial }{{\partial y}}\left( {{e^{xy}}} \right)j \\&= {e^{xy}}\left( {1 + xy} \right)i + \left( {{x^2}{e^{xy}}}\right)j \\\end{aligned}