Find the divergence of the vector field. F = 10x^6 i - xy^5 j.

Question:

Find the divergence of the vector field.

{eq}F = 10x^6 i - xy^5 j {/eq}

Divergence:

The divergence of a vector field is a scalar. It is defined for both two-dimensional and three-dimensional vector-fields. The divergence of a two-dimensional vector field {eq}\vec{F} = F_1 \vec{i} + F_2 \vec{j} {/eq} is defined by using the dot product:

$$\nabla \cdot \vec{F} $$

Here: {eq}\nabla = \dfrac{\partial}{\partial x} \vec{i}+\dfrac{\partial}{\partial y} \vec{j} {/eq}

Answer and Explanation:

The given vector field is:

$$\vec{F}=10 x^{6} \vec{i}-x y^{5} \vec{j} $$

The divergence is given by the dot product:

$$\nabla \cdot \vec{F} $$

Here: {eq}\nabla = \dfrac{\partial}{\partial x} \vec{i}+\dfrac{\partial}{\partial y} \vec{j} {/eq}

So we get:

$$\begin{align} \text{Divergence} &= \nabla \cdot \vec{F}\\ & = ( \dfrac{\partial}{\partial x} \vec{i}+\dfrac{\partial}{\partial y} \vec{j}) \cdot (10 x^{6} \vec{i}-x y^{5} \vec{j}) \\ & = \dfrac{\partial}{\partial x}\left(10 x^{6}\right) + \dfrac{\partial}{\partial y}\left(-x y^{5}\right) \\ &=60 x^{5}-5xy^4 \end{align} $$


Therefore, the divergence of the given vector field = {eq}\boxed{\mathbf{60 x^{5}-5xy^4}} {/eq}


Learn more about this topic:

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The Dot Product and Vectors: Definition & Formula

from Physics: High School

Chapter 2 / Lesson 11
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