Evaluate \nabla x ( \frac{r}{r^2}).

Question:

Evaluate {eq}\nabla x ( \frac{r}{r^2}). {/eq}

Vector Field:

{eq}\\ {/eq}

For a smooth closed surface in 3-D space , where {eq}r {/eq} is the distance between origin and point {eq}(x,y,z) {/eq}, we can compute {eq}\nabla ( \frac{1}{r}) {/eq} as {eq}\nabla ( \frac{1}{r})=\dfrac{\delta \frac{1}{r} }{\delta x}+\dfrac{\delta \frac{1}{r} }{\delta y}+\dfrac{\delta \frac{1}{r} }{\delta z}. {/eq}. The gradient o a vector field is the collection of partial derivatives of its component and if we are told to find the gradient of a particular component of that vector field then we can compute it by finding the partial derivative of that component. The gradient vectors are generally perpendicular to contour lines of that vector valued function.

Answer and Explanation:

{eq}\\ {/eq}

Let there be a smooth closed surface {eq}S {/eq} in 3-Dimensional space and {eq}r {/eq} be the distance between origin and point {eq}(x,y,z) {/eq}.

So, {eq}\nabla _x ( \frac{r}{r^2})=\nabla_x ( \frac{1}{r}) {/eq}

{eq}\Rightarrow \boxed{\nabla_x ( \frac{1}{r})=\frac{\delta \frac{1}{r} }{\delta x}} {/eq}

Hence, by this formula, we can calculate {eq}\nabla_x ( \frac{r}{r^2}). {/eq}.


Learn more about this topic:

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Vectors: Definition, Types & Examples

from Common Entrance Test (CET): Study Guide & Syllabus

Chapter 57 / Lesson 3
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