Vector Equation | Proving ∇×(a × b)

Thread starter joris_pixie
Start date Nov 21, 2009
Tags

Vector

In summary, a vector equation is an equation involving vectors that is used to describe physical quantities. To prove ∇×(a × b), one must use the properties of vector calculus. The significance of this operation is that it describes the rotation of a vector field and is commonly used in physics and engineering. An example of its use in real life is in fluid dynamics to calculate vorticity. Common mistakes when working with ∇×(a × b) include not understanding vector calculus, not properly applying operations, and not recognizing when the equation is equivalent to zero. It is important to have a strong understanding and carefully follow the steps when working with this equation.

Nov 21, 2009

joris_pixie

vector equation [SOLVED]

Hello, I'm trying to proof that
∇×(a × b) = a(∇ · b) − b(∇ · a) + (b · ∇)a − (a · ∇)b
(where a & b are vectors)

But I'm stuck...

Probably because this isn't correct:

∇×(a × b) = a(∇b) - b(∇a) ?

But i don't know why!

Could somebody please help !
Kind regards

Last edited: Nov 21, 2009

Physics news on Phys.org

Nov 21, 2009

joris_pixie

whoops, already get it !

1. What is a vector equation?

A vector equation is an equation that involves vectors, which are mathematical quantities that have both magnitude and direction. It is used to describe physical quantities such as force, velocity, and acceleration.

2. How do you prove ∇×(a × b)?

In order to prove ∇×(a × b), you need to use the properties of vector calculus, specifically the cross product and the gradient operator. By applying these properties, you can manipulate the equation to show that it is equivalent to zero, thus proving its validity.

3. What is the significance of ∇×(a × b)?

∇×(a × b) is a vector operation known as the curl, which describes the rotation of a vector field. It is significant in physics and engineering as it is used to calculate the amount of circulation or rotation in a fluid or electromagnetic field.

4. Can you provide an example of using ∇×(a × b) in real life?

One example of using ∇×(a × b) in real life is in fluid dynamics. In this context, the curl can be used to calculate the vorticity of a fluid flow, which is the tendency of a fluid to rotate. This is important in understanding and predicting the behavior of fluids in various situations, such as in weather patterns or aircraft design.

5. What are some common mistakes when working with ∇×(a × b)?

Some common mistakes when working with ∇×(a × b) include not understanding the properties of vector calculus, not properly applying the cross product and gradient operations, and not recognizing when the equation is equivalent to zero. It is important to have a strong understanding of vector calculus and to carefully follow the steps when manipulating the equation.