Chapter 14: Q 26. (page 1154)

Find the divergence and curl of the following vector fields.
$F (x, y) = x^{2} y i + x y^{3} j$

Short Answer

Expert verified

$\nabla \cdot F (x, y) = 2 x y + 3 x y^{2}$

$\nabla \times F (x, y) = (y^{3} - x^{2}) k$

Step by step solution

Given Information

The given expression is $F (x, y) = x^{2} y i + x y^{3} j$

Step 2. Finding the Divergence and Curl

Find the divergence of the given vector function as shown.

F (x, y) = x^{2} y i + x y^{3} j

localid="1651905655379" role="math" $\nabla \cdot F (x, y) = \frac{\partial}{\partial x} F_{1} + \frac{\partial}{\partial y} F_{2}$

Evaluating the divergence

localid="1651905645792" role="math" $\nabla \cdot F (x, y) = 2 x y + 3 x y^{2}$

Find the curl of the given vector function as shown.

$\nabla \times F (x, y) = (\frac{\partial}{\partial y} F_{3} - \frac{\partial}{\partial z} F_{2}) i + (\frac{\partial}{\partial z} F_{1} - \frac{\partial}{\partial x} F_{3}) j + (\frac{\partial}{\partial x} G_{2} - \frac{\partial}{\partial y} G_{1}) k$

Evaluating the curl

localid="1651525649420" $\nabla \times F (x, y) = 0 i + 0 j + (y^{3} - x^{2}) k$

$\nabla \times F (x, y) = (y^{3} - x^{2}) k$

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Find the divergence and curl of the following vector fields.
$F (x, y) = x^{2} y i + x y^{3} j$

Short Answer

Step by step solution

Given Information

Step 2. Finding the Divergence and Curl

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Find the divergence and curl of the following vector fields.Fx,y=x2yi+xy3j

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Step by step solution

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Step 2. Finding the Divergence and Curl

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Find the divergence and curl of the following vector fields.
$F (x, y) = x^{2} y i + x y^{3} j$