Vector Calculus Identity proof?

In summary, the identity in question is a variation of a well-known E&M vector identity, which can be found in various E&M textbooks including J.D. Jackson's. It is not listed on Wiki's list of vector calculus identities, but can be found in equations 1.2 to 1.4 in the book on Turbulence provided in the link. The identity involves the vector ## \omega ##, defined as ## \omega=\nabla \times u ##, and follows the form of ## \nabla (a \cdot b)=a \cdot \nabla b +b \cdot \nabla a + \ a \times \nabla \times b + \ b \times
  • #1
26
3
The following identity is found in a book on Turbulence:

upload_2016-6-15_19-33-17.png


Can someone provide a proof of this identity? It isn't listed in the list of vector calculus identities on Wiki.

Thanks
 
Physics news on Phys.org
  • #3
Apparently ## \omega ## is defined as ## \omega=\nabla \times u ##. This one is actually a variation of a well-known E&M vector identity (i.e. a vector calculus identity used quite often in E&M coursework): ## \nabla (a \cdot b)=a \cdot \nabla b +b \cdot \nabla a + \ a \times \nabla \times b + \ b \times \nabla \times a ## ## \ ## where ## a=b=u ##. (This one is found on the cover of J.D. Jackson's E&M textbook as well as other E&M textbooks.)
 
Last edited:
  • Like
Likes FluidStu

What is a vector calculus identity?

A vector calculus identity is a mathematical equation that relates different operations on vectors, such as dot product, cross product, and vector differentiation, to each other. These identities are used to simplify and solve problems in vector calculus.

Why are vector calculus identities important?

Vector calculus identities are important because they allow us to manipulate and solve complex vector equations and problems more easily. They also provide a deeper understanding of the relationships between different vector operations.

What is the process for proving a vector calculus identity?

The process for proving a vector calculus identity involves using basic vector properties and operations, such as the distributive and associative properties, along with the properties of dot and cross products. By manipulating the equations using these properties, we can show that both sides of the identity are equivalent.

What are some common vector calculus identities?

Some common vector calculus identities include the dot product identity (a · (b x c) = (a x b) · c), the cross product identity (a x (b x c) = (a · c)b - (a · b)c), and the triple product identity (a x (b x c) = b(a · c) - c(a · b)).

How can I apply vector calculus identities in real-world situations?

Vector calculus identities can be applied in many fields of science and engineering, including physics, engineering, and computer graphics. They can be used to solve problems involving forces, motion, and geometric transformations. They are also essential in understanding and solving problems in electromagnetic theory and fluid mechanics.

Suggested for: Vector Calculus Identity proof?

Replies
5
Views
2K
Replies
3
Views
1K
Replies
2
Views
983
Replies
3
Views
908
Replies
5
Views
1K
Replies
6
Views
962
Replies
1
Views
671
Replies
9
Views
972
Back
Top