Question about divergence and curl:

Thread starter Aronyak
Start date Jul 16, 2012
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Curl Divergence

In summary, divergence and curl are mathematical operations used to analyze vector fields. Divergence measures expansion or contraction, while curl measures rotation. They have physical significance in fluid dynamics and electromagnetism, and are calculated using the dot and cross products with the "del" operator. Their properties include linearity, the product rule, and being conservative. They are related through the fundamental theorem of calculus for line integrals.

Jul 16, 2012

Aronyak

Please Someone explain why:
1.div(F×G)=GcurlF-FcurlG
2.curl(F×G)=F.divG-G.divF+(G.∇).F-(F.∇).G

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Jul 16, 2012

tiny-tim

Science Advisor

Homework Helper

Welcome to PF!

Hi Aronyak! Welcome to PF!

You can work it out, either by using x y and z coordinates, or by using the einstein summation convention …

show us what you get.

What is divergence and curl?

Divergence and curl are two mathematical operations that are used to analyze vector fields. Divergence measures the rate at which a vector field is expanding or contracting at a given point, while curl measures the rate at which it is rotating around that point.

What is the physical significance of divergence and curl?

Divergence and curl have physical significance in various fields of science and engineering. Divergence is important in fluid dynamics, as it helps determine the flow of fluids. Curl is significant in electromagnetism, as it helps describe the magnetic field around a current-carrying wire.

How do you calculate divergence and curl?

Divergence is calculated by taking the dot product of the vector field with the operator "del" (∇). Curl is calculated by taking the cross product of the vector field with the operator "del" (∇).

What are the properties of divergence and curl?

The properties of divergence and curl include linearity, meaning they follow the rules of addition and scalar multiplication. They also satisfy the product rule, meaning the divergence or curl of a product is equal to the product of the individual divergences or curls. Additionally, they have the property of being conservative, meaning their path integrals are independent of the path taken.

How are divergence and curl related?

Divergence and curl are related through the fundamental theorem of calculus for line integrals. The divergence of a vector field is equal to the flux of its curl through any closed surface, and the curl of a vector field is equal to the line integral of its divergence around any closed curve.