Chapter 14: Q 28. (page 1154)

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

Short Answer

Expert verified

$\nabla \cdot F (x, y, z) = 0$

$\nabla \times F (x, y, z) = (- 2 y - 2 z) i + (- 2 z - 2 x) j + (- 2 x - 2 y) k$

Step by step solution

Given Information

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

Finding the Divergence and Curl

Find the divergence of the given vector function as shown.

$\nabla \cdot F (x, y, z) = \frac{\partial}{\partial x} F_{1} + \frac{\partial}{\partial y} F_{2} + \frac{\partial}{\partial z} F_{3}$

Evaluating the divergence

$\nabla \cdot F (x, y, z) = 0 + 0 + 0 = 0$

Find the curl of the given vector function as shown.

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

Evaluating the curl

$= (- 2 y - 2 z) i + (- 2 z - 2 x) j + (- 2 x - 2 y) k$

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

Short Answer

Step by step solution

Given Information

Finding the Divergence and Curl

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Find the divergence and curl of the following vector fields.Fx,y,z=y2-z2i+z2-x2j+x2-y2k

Short Answer

Step by step solution

Given Information

Finding the Divergence and Curl

One App. One Place for Learning.

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