Hi, can someone give me some assistance with the following questions?(adsbygoogle = window.adsbygoogle || []).push({});

1. Let f(x,y,z), g(x,y,z) and h(x,y,z) be any C^2 scalar functions. Prove that [tex]\nabla \bullet \left( {f\nabla g \times \nabla h} \right) = \nabla f \bullet \left( {\nabla g \times \nabla h} \right)[/tex] .

2. Let S be the part of the ellipsoid 2x^2 + y^2 + (z-1)^2 = 5 for z <=0 and [tex]\mathop F\limits^ \to = \left( {e^{y + z} + 3y,xe^{y + z} ,\cos \left( {xyz} \right) + z^3 } \right)[/tex] .

Evaluate [tex]\int\limits_{}^{} {\int\limits_S^{} {\left( {\nabla \times \mathop F\limits^ \to } \right)} \bullet d\mathop S\limits^ \to } [/tex].

(Use the normal to the surface pointing downwards.)

My working:

1. I will use [tex]

\nabla \bullet \left( {\mathop A\limits^ \to \times \mathop B\limits^ \to } \right) = \mathop B\limits^ \to \bullet \left( {\nabla \times \mathop A\limits^ \to } \right) - \mathop A\limits^ \to \left( {\nabla \times \mathop B\limits^ \to } \right)

[/tex] and [tex]\nabla \times \left( {f\mathop F\limits^ \to } \right) = f\nabla \times \mathop F\limits^ \to + \nabla f \times \mathop F\limits^ \to [/tex].

[tex]

\nabla \bullet \left( {f\nabla g \times \nabla h} \right)

[/tex]

[tex]

= \nabla h \bullet \left( {\nabla \times f\nabla g} \right) - f\nabla g \bullet \left( {\nabla \times \nabla h} \right)

[/tex]...from the identies above .The second bracket is zero since curl(grad(h)) = 0 vector.

[tex]

= \nabla h \bullet \left( {\nabla \times f\nabla g} \right)

[/tex]

[tex]

= \nabla h \bullet \left( {f\nabla \times \nabla g + \nabla f \times \nabla g} \right)

[/tex]...using identities listed above

[tex]

= \nabla h \bullet \left( {\nabla f \times \nabla g} \right)

[/tex] since curl(grad(g)) = 0 vector.

This is as far as I get in the first question.

For the surface integral I calculated [tex]\nabla \times \mathop F\limits^ \to = \left( { - xz\sin \left( {xyz} \right) - xe^{y + z} ,e^{y + z} + yz\sin \left( {xyz} \right), - 3} \right)[/tex].

I would normally parameterise the surface to find a normal to the surface. I'm not sure how to that or if I need to do that in this question.

Any help would be good thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: A vector identity and surface integral

**Physics Forums | Science Articles, Homework Help, Discussion**