Another mathamatician neede to check this CURL-GRAD-DIV

[andyc10]

Homework Statement

I have repeated this problem over and over again so either the question is wrong or my likely its my method that's at fault. Please can someone check this for me.

Prove this is true:
$$\nabla\times(\nabla \times F)=-\nabla^{2}F+\nabla(\nabla.F)$$

for $$F=x^{2}z^{3}i$$

The Attempt at a Solution

Im getting for the LHS:

$$(\nabla \times F)= 3x^{2}z^{3} j$$

$$\nabla\times(\nabla \times F)= -6x^{2}z i + 6xz^{2}k$$

and for the right:

$$-\nabla^{2}F=-2z^{3} i$$

$$(\nabla.F)=2xz^{3}$$

$$\nabla(\nabla.F)=2z^{3}i + 6xz^2k$$

$$-\nabla^{2}F+\nabla(\nabla.F)=6xz^{2} k$$

$$\Rightarrow \nabla\times(\nabla \times F)\neq-\nabla^{2}F+\nabla(\nabla.F)$$
for $$F=x^{2}z^{3}i$$

 

[Dvsdvs]

-\nabla^{2}F=-2z^{3} i[/tex]

$$(\nabla.F)=2xz^{3}$$

$$\nabla(\nabla.F)=2z^{3}i + 6xz^2k$$

$$-\nabla^{2}F+\nabla(\nabla.F)=6xz^{2} k$$

$$\Rightarrow \nabla\times(\nabla \times F)\neq-\nabla^{2}F+\nabla(\nabla.F)$$
for $$F=x^{2}z^{3}i$$

isn't $$\nabla$$.$$\nabla$$ = $$\nabla$$^2= d^2/dx^2+d^2/dy^2+etc.

ugh the text never comes out right why is line 1 not equal to negative line 3?

 

[slider142]

and for the right:

$$-\nabla^{2}F=-2z^{3} i$$

This is not correct.

 

[Cyosis]

It seems you've taken the scalar Laplacian and then made a vector out of it. Make sure you use the vector Laplacian and you will get the extra term you need to proof the equality.

 

[Vid]

This question doesn't make any sense. Using the vector laplacian is trivial since it's defined by using the other two terms in your equations.

 

[cristo]

This question doesn't make any sense. Using the vector laplacian is trivial since it's defined by using the other two terms in your equations.

Presumably, given that we are in Cartesian coordinates, you use the fact that $$\nabla^2\vec{F}=\nabla^2F_x\mathbf{i}+\nabla^2F_y\mathbf{j}+\nabla^2F_z\mathbf{k}$$

 

[andyc10]

This is not correct.

http://en.wikipedia.org/wiki/Vector_Laplacian

$$A_{x}=x^{2}z^{3}$$

$$A_{y}=0$$

$$A_{z}=0$$


$$\nabla^{2}A_{x}=2z^{3}$$

$$\Rightarrow\nabla^{2}A=2z^{3}i$$

$$\Rightarrow-\nabla^{2}F=-2z^{3}i$$

is this not right?

 

[cristo]

http://en.wikipedia.org/wiki/Vector_Laplacian

$$A_{x}=x^{2}z^{3}$$

$$A_{y}=0$$

$$A_{z}=0$$


$$\nabla^{2}A_{x}=2z^{3}$$

This isn't correct. $$\nabla^2 F_x=\nabla^2x^2z^3=2z^3+6zx^2$$

 

[Cyosis]

The normal Laplacian is given by $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$. You already stated that $A_x=x^2 z^2$. Now all you have to do is apply the normal Laplacian to A_x.

 

[andyc10]

Thanks guys, sorted it.