Vector calc identities

In summary, the conversation is about deriving the vector calculus "del" relationships and whether there is a method similar to using trig identities. The suggestion is to use x,y,z components for the vector and rearrangements to prove the relationships, which can be easily done using indices. The formula for ##\vec{\nabla} \times (\vec{\nabla} \times \vec{\xi})## is given as an example.
  • #1

member 428835

hey all!

i was hoping someone could either state an article or share some knowledge with a way (if any) to derive the vector calculus "del" relationships. i.e. $$ \nabla \cdot ( \rho \vec{V}) = \rho (\nabla \cdot \vec{V}) + \vec{V} \cdot (\nabla\rho)$$

now i do understand this to be like the product rule but is there any way to derive this in a similar fashion to other vector calculus identities, perhaps using the Kronecker delta or the permutation epsilon?

as an example, i can derive almost any trig identity using $$e^{i \theta}=\cos \theta+i \sin \theta$$ by making changes to [itex]\theta[/itex], perhaps letting [itex]\theta = \alpha + \beta[/itex] and then equating real and imaginary parts and doing a little bit of algebra.

is there anything like this for vector identities (perhaps not a base equation, but a method?)

thanks for your help!
 
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  • #2
If you use x,y,z components for the vector then do some rearrangements you can prove it.
 
  • #3
Pretty much every curl, divergence, and gradient identity can be proven easily and elegantly using indices. For example, consider the very useful formula ##\vec{\nabla} \times (\vec{\nabla} \times \vec{\xi}) = \vec{\nabla}(\vec{\nabla}\cdot \vec{\xi}) - \vec{\nabla}^2 \vec{\xi}##.

We have ##(\vec{\nabla} \times (\vec{\nabla} \times \vec{\xi}))^i \\= \epsilon^{ijk}\partial_{j}(\epsilon_{krl}\partial^{r}\xi^{l}) \\= (\delta^{i}_{r}\delta^{j}_{l} - \delta^{j}_{r}\delta^{i}_{l})\partial_{j}\partial^{r}\xi^{l} \\= \partial^{i}\partial_{l}\xi^{l} - \partial_{r}\partial^{r}\xi^{i}\\ = (\vec{\nabla}(\vec{\nabla}\cdot \vec{\xi}))^i - (\vec{\nabla}^2\vec{\xi})^i##
as desired.
 
  • #4
thanks wannabe Newton
 

What are vector calc identities?

Vector calc identities are mathematical equations that relate different vector quantities such as position, velocity, and acceleration. They are used to solve problems in vector calculus and are fundamental to understanding the behavior of vector quantities in three-dimensional space.

What is the difference between a scalar and a vector?

A scalar is a quantity that has only magnitude, while a vector has both magnitude and direction. Scalars are represented by a single number, while vectors are represented by both magnitude and direction, often using arrows.

How are vector calc identities used in physics?

Vector calc identities are used in physics to describe the motion of objects in space. They are essential in calculating the velocity and acceleration of objects, as well as the forces acting on them. They are also used in fields such as electromagnetism and fluid dynamics.

What are some common vector calc identities?

Some common vector calc identities include the dot product, cross product, and vector triple product. The dot product calculates the scalar projection of one vector onto another, the cross product calculates the vector perpendicular to two given vectors, and the vector triple product calculates the volume of a parallelepiped formed by three given vectors.

How can I apply vector calc identities to real-world problems?

Vector calc identities can be applied to real-world problems such as calculating the trajectory of a projectile, determining the forces acting on a moving object, or analyzing the flow of fluids. They are also used in engineering and design to optimize structures and systems.

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