Find the divergence of a vector $\vec{A} = x^{2} {\hat{a}}_{x} + 6 y^{2} {\hat{a}}_{y} + z^{3} {\hat{a}}_{z}$ at point P (2, 4, 1).

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Find the divergence of a vector $\vec{A} = x^{2} {\hat{a}}_{x} + 6 y^{2} {\hat{a}}_{y} + z^{3} {\hat{a}}_{z}$ at point P (2, 4, 1).

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55

Answer 1

Concept:

For a vector F = F1i + F2j + F3k

$D i v = \nabla . F = \frac{\partial F_{1}}{\partial x} + \frac{\partial F_{2}}{\partial y} + \frac{\partial F_{3}}{\partial z}$

$C u r l = \nabla \times F = | \begin{matrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_{1} & F_{2} & F_{3} \end{matrix} |$

Calculation:

$\vec{A} = x^{2} {\hat{a}}_{x} + 6 y^{2} {\hat{a}}_{y} + z^{3} {\hat{a}}_{z}$

The divergence is given by,

$\nabla . \vec{A} = \frac{\partial}{\partial x} (x^{2}) + \frac{\partial}{\partial y} (6 y^{2}) + \frac{\partial}{\partial z} (z^{3})$

= 2x + 12y + 3z²

At the given point P (2, 4, 1), divergence is:

= 2(2) + 12(4) + 3(1) = 55

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Question

Find the divergence of a vector $\vec{A} = x^{2} {\hat{a}}_{x} + 6 y^{2} {\hat{a}}_{y} + z^{3} {\hat{a}}_{z}$ at point P (2, 4, 1).

Answer (Detailed Solution Below)

Detailed Solution

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Question

Find the divergence of a vector A→=x2a^x+6y2a^y+z3a^z at point P (2, 4, 1).

Answer (Detailed Solution Below)

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Find the divergence of a vector $\vec{A} = x^{2} {\hat{a}}_{x} + 6 y^{2} {\hat{a}}_{y} + z^{3} {\hat{a}}_{z}$ at point P (2, 4, 1).