# Vector Cross Product With Its Curl

• John Delaney
In summary, when you are working with index notations, it is important to keep track of all terms and their possible duplicates in order to arrive at the correct result.
John Delaney
Homework Statement
Prove A X (∇ X A) = ∇(A²/2) - A · ∇A
Relevant Equations
εijk εlmk = δil δjm - δim δjl
Starting with LHS:

i εijk Aj (∇xA)k

i εijk εlmk Aj (d/dxl) Am

il δjm - δim δjl) Aj (d/dxl) Ami

δil δjm Aj (d/dxl) Ami - δim δjl Aj (d/dxl) Ami

Aj (d/dxi) Aji - Aj (d/dxj) Aii

At this point, the LHS should equal the RHS in the problem statement, but I have no clue where the 1/2 comes from...been trying to figure it out for the past few hours at this point. Rather, I get this and I'm not sure where my lapse in understanding is (I'm relatively new to index notation):

∇(A²) - A · ∇A

Delta2
What is the derivative of f(x)=x2/2?

Notice that ##A^2 = A \cdot A##. When the gradient operator is applied to this term, you get two terms looking like this: ##A(\nabla A)##. In your initial work, you only have one of these terms, so to account for the duplicate, you must divide ##\nabla A^2## by two, which results in the term ##\nabla \frac{A^2}{2}## appearing in the identity you are attempting to prove.

Delta2

## What is a vector cross product?

A vector cross product, also known as a cross product, is a mathematical operation between two vectors that results in a third vector that is perpendicular to both of the original vectors.

## How is a vector cross product calculated?

A vector cross product can be calculated using the right-hand rule, which involves taking the cross product of the unit vectors of the two original vectors and then multiplying them together. The resulting vector is the cross product.

## What is the geometric interpretation of a vector cross product?

The geometric interpretation of a vector cross product is that it represents the area of a parallelogram formed by the two original vectors. The direction of the resulting vector is perpendicular to the plane of the parallelogram.

## What is the relationship between a vector cross product and its curl?

The curl of a vector is equal to the vector cross product of its gradient and the original vector. This relationship is used in vector calculus to solve problems involving vector fields.

## What are some real-world applications of vector cross product and its curl?

Vector cross products and their curl have many applications in physics and engineering, such as calculating the torque on an object, determining the direction of magnetic fields, and solving fluid mechanics problems.

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