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Hi, can someone give me some assistance with the following questions?

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.

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.

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