Checking $\nabla x (uv(hat))$ Equation

[squenshl]

How do I check $$\nabla$$ x (uv(hat)) = ($$\nabla$$u) x v(hat) + u($$\nabla$$ x v(hat)).

 

[LCKurtz]

How do I check $$\nabla$$ x (uv(hat)) = ($$\nabla$$u) x v(hat) + u($$\nabla$$ x v(hat)).

Changing your notation a little, if f(x,y,z) is a scalar and

V(x,y,z) = <u(x,y,z),v(x,y,z),w(x,y,z)> is a vector, to show

$$\nabla \times f\vec V = \nabla f \times\vec V + f\nabla \times \vec V$$

just work both sides out in terms of components and compare. I don't think it matters whether V is a unit vector.

 

[gabbagabbahey]

The same way you prove every other vector calculus identity... express everything in terms of components, calculate the derivatives and simplify.

Even if this isn't an assigned homework problem, it's still a homework type problem and you should follow the homework template.

 

[squenshl]

What if I wanted to use Cartesian coordinates, the definitions of the determinant, gradient and cross product of 2 vectors.

 

[Redbelly98]

What if I wanted to use Cartesian coordinates, the definitions of the determinant, gradient and cross product of 2 vectors.

As a matter of fact, you pretty much have to use all of those. And work things out in terms of components of vectors, as others have said.

Is this homework?

 

[squenshl]

Na. Studying for a test. It was a on a practice test.

 

[Redbelly98]

Na. Studying for a test. It was a on a practice test.

Moderator's note:

I have moved this thread to the Homework & Coursework Questions area. We have guidelines on what belongs there, and this definitely qualifies.

At this point, normal rules for Homework & Coursework apply. The OP should show an attempt at solving the problem before further help is given.