Chapter 1: Q15P (page 18)

Calculate the divergence of the following vector functions:
$(a) v_{a} = x^{2} \hat{x} + 3 x z^{2} \hat{y} - 2 x z \hat{z} (b) v_{b} = x y \hat{x} + 2 y z \hat{y} + 3 z x \hat{z} (c) v_{c} = y^{2} \hat{x} + (2 x y + z^{2}) + 2 y z \hat{z}$

Short Answer

Expert verified

(a) The divergence is 0.

(b) The divergence is $^{3 x + y + 2 z}$ .

Step by step solution

Write the expression for the divergence of the function.

Determine the divergence of the function.

$\nabla . v = (\hat{x} \frac{\partial}{\partial x} + \hat{y} \frac{\partial}{\partial y} + \hat{z} \frac{\partial}{\partial z}) (v_{x} \hat{x} + v_{y} \hat{y} + v_{z} \hat{z}) = \frac{\partial v_{x}}{\partial x} + \hat{y} \frac{\partial v_{y}}{\partial y} + \hat{z} \frac{\partial v_{z}}{\partial z}$

Solve for the divergence of part (a).

Consider the given vector is,

$v_{a} = x^{2} \hat{x} + 3 x z^{2} y - 2 x z \hat{z}$

Solve for the divergence.

$\nabla . v_{a} = \frac{\partial}{\partial x} x^{2} + \frac{\partial}{\partial y} 3 x z^{2} - \frac{\partial}{\partial z} 2 x z = 2 x + 0 - 2 x = 0$

Therefore, the divergence is 0.

Solve for the divergence of part (b).

Consider the given vector is,

$v_{b} = x y \hat{x} + 2 y z^{2} y + 3 z x \hat{z}$

Solve for the divergence.

$\nabla . v_{a} = \frac{\partial}{\partial x} x y + \frac{\partial}{\partial y} 2 y z - \frac{\partial}{\partial z} x z = 3 x + y + 2 z$

Therefore, the divergence is $^{3 x + y + 2 z}$

Solve for the divergence of part (c).

Consider the given vector is,

$v_{a} = y^{2} \hat{x} + (2 x y + z^{2}) + 2 y z \hat{z}$

Solve for the divergence.

$\nabla . v_{a} = \frac{\partial}{\partial x} y^{2} + \frac{\partial}{\partial y} (2 x y + z^{2}) + \frac{\partial}{\partial z} 2 y z = 0 + (2 x + 0) + 2 y = 2 (x + y)$

Therefore, the divergence is $^{2 (x + y)}$ .

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Calculate the divergence of the following vector functions:
$(a) v_{a} = x^{2} \hat{x} + 3 x z^{2} \hat{y} - 2 x z \hat{z} (b) v_{b} = x y \hat{x} + 2 y z \hat{y} + 3 z x \hat{z} (c) v_{c} = y^{2} \hat{x} + (2 x y + z^{2}) + 2 y z \hat{z}$

Short Answer

Step by step solution

Write the expression for the divergence of the function.

Solve for the divergence of part (a).

Solve for the divergence of part (b).

Solve for the divergence of part (c).

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Calculate the divergence of the following vector functions:(a)va=x2x^+3xz2y^-2xzz^(b)vb=xyx^+2yzy^+3zxz^(c)vc=y2x^+(2xy+z2)+2yzz^

Short Answer

Step by step solution

Write the expression for the divergence of the function.

Solve for the divergence of part (a).

Solve for the divergence of part (b).

Solve for the divergence of part (c).

One App. One Place for Learning.

Most popular questions from this chapter

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Calculate the divergence of the following vector functions:
$(a) v_{a} = x^{2} \hat{x} + 3 x z^{2} \hat{y} - 2 x z \hat{z} (b) v_{b} = x y \hat{x} + 2 y z \hat{y} + 3 z x \hat{z} (c) v_{c} = y^{2} \hat{x} + (2 x y + z^{2}) + 2 y z \hat{z}$