# Proof - Vector Calculus - Curl

• cristina89
In summary, curl in vector calculus is a mathematical operation that describes the rotation or rotational tendency of a vector field at a given point. It is calculated using the partial derivatives of a vector field with respect to each coordinate axis, and a high curl value signifies a strong rotational tendency of the vector field. In physics, curl is used to describe the movement of fluids and electromagnetic fields, as well as the rotation of rigid bodies. It is different from divergence, which measures the expansion or contraction of a vector field at a given point.
cristina89
I need to prove this: u x ($\nabla$ x u) = $\frac{1}{2}$$\nabla$(u²) - (u $\cdot$ $\nabla$)u.

I've came to this: uj∂iuj - uj∂jui (i think it's correct)
But how this 1/2 appears?

welcome to pf!

hi cristina89! welcome to pf!

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cristina89 said:
I need to prove this: u x ($\nabla$ x u) = $\frac{1}{2}$$\nabla$(u²) - (u $\cdot$ $\nabla$)u.

I've came to this: uj∂iuj - uj∂jui (i think it's correct)
But how this 1/2 appears?

'cos ∂i(ujuj)= 2uji(uj)

## 1. What is curl in vector calculus?

Curl is a mathematical operation that describes the rotation or rotational tendency of a vector field at a given point. It is represented by a vector quantity and is used to analyze the behavior of the vector field.

## 2. How is curl calculated?

Curl is calculated using the partial derivatives of a vector field with respect to each coordinate axis. The resulting vector is the curl of the original vector field.

## 3. What does a high curl value signify?

A high curl value indicates a strong rotational tendency of the vector field at a given point. This means that the vector field is changing direction rapidly around that point.

## 4. What is the significance of curl in physics?

Curl is used in physics to describe the movement of fluids and electromagnetic fields. It is also used in mechanics to analyze the rotation of rigid bodies.

## 5. How is curl different from divergence?

Curl and divergence are both mathematical operations used in vector calculus, but they have different meanings and calculations. While curl measures the rotational tendency of a vector field, divergence measures the expansion or contraction of the field at a given point.

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