# Help deriving Lagrange's Formula with the levi-civita symbol

In summary, the conversation discusses the use of determinants and the Levi-Civita symbol to prove a formula involving cross products and dot products. The final formula is given and the use of \cdot instead of \bullet is suggested.
Ok, so I'm really at a loss as to how to do this. I can prove the formula by just using determinants, but I don't really know how to do manipulations with the levi-civita symbol.
Here's what I have so far:
$$(\vec{B} \times \vec{C})_{i} = \epsilon_{ijk}(B_{j}C_{k})\vec{e_{i}}$$

And I'm trying to get to:
$$\vec{A} \times (\vec{B} \times \vec{C}) = B(A \bullet C) - C(A \bullet B)$$

Does anyone have any suggestions?
Thanks

Ok So I figured it out, I'll just post the answer for the sake of completeness:

$$(\vec{B} \times \vec{C})_{k} = \epsilon_{kmn} (B_{m} C_{n})$$

$$let (\vec{B} \times \vec{C}) = \vec{N}$$

$$\vec{A} \times (\vec{B} \times \vec{C}) = \vec{A} \times \vec{N}$$
$$(\vec{A} \times \vec{N})_{i} = \epsilon_{ijk} A_{j} N_{k}$$
$$= \epsilon_{ijk} A_{j} (\epsilon_{kmn} B_{m} C_{n})$$
$$= \epsilon_{ijk} \epsilon_{kmn} (A_{j} B_{m} C_{n})$$

$$\epsilon_{ijk} \epsilon_{mnk} = \delta_{im} \delta_{jn} - \delta_{in} \delta_{jm}$$

$$\vec{A} \times (\vec{B} \times \vec{C}) = (\delta_{im} \delta_{jn} - \delta_{in} \delta_{jm}) A_{j} B_{m} C_{n}$$
$$= B_{i} A_{j} C_{j} - A_{j} B_{j} C_{i}$$
$$= \vec{B}(\vec{A} \bullet \vec{C}) - \vec{C}(\vec{A} \bullet \vec{B})$$

$$(\vec{B} \times \vec{C})_{i} = \epsilon_{ijk}(B_{j}C_{k})\vec{e_{i}}$$

Should be:

$$(\vec{B} \times \vec{C})_{i} = \epsilon_{ijk}(B_{j}C_{k})$$

And start using \cdot instead of that big black ball :-D

Haha wow this seems like so long ago. I couldn't find the dot for dot product, so thanks for that :D

## What is Lagrange's Formula?

Lagrange's Formula is a mathematical equation used in vector calculus that expresses the derivative of a vector function with respect to a given variable as a linear combination of the partial derivatives of the components of the vector function.

## What is the Levi-Civita symbol?

The Levi-Civita symbol, also known as the permutation symbol, is a mathematical symbol used to represent the sign of a permutation of a set of indices. It is commonly used in vector calculus and differential geometry.

## How is the Levi-Civita symbol related to Lagrange's Formula?

The Levi-Civita symbol is used in Lagrange's Formula to express the derivative of a vector function in terms of the components of the vector function and their partial derivatives. It helps to simplify the calculation and makes the formula more concise.

## What are the steps for deriving Lagrange's Formula with the Levi-Civita symbol?

The steps for deriving Lagrange's Formula with the Levi-Civita symbol involve setting up the formula using the vector function and its components, using the properties of the Levi-Civita symbol to simplify the calculation, and then rearranging the terms to get the final result.

## What are some applications of Lagrange's Formula with the Levi-Civita symbol?

Lagrange's Formula with the Levi-Civita symbol is commonly used in physics and engineering to solve problems related to vector fields, such as calculating the curl and divergence of a vector field. It is also used in the study of fluid dynamics, electromagnetism, and general relativity.

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