Does this make any sense at all? Surface Area/Integral

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SUMMARY

The discussion centers on the calculation of surface integrals using parametric surfaces defined by the vector function \(\mathbf{r}(u,v) = \). Participants confirm that the differential area element \(dS\) can indeed be expressed as \(dS = |\mathbf{r_u} \times \mathbf{r_v}|dA\), where \(\mathbf{r_u}\) and \(\mathbf{r_v}\) are the partial derivatives of the parametric surface with respect to \(u\) and \(v\), respectively. This formulation is essential for evaluating surface integrals of the form \(\iint f(x,y,z)\;dS\).

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  • Knowledge of vector calculus, specifically cross products
  • Familiarity with surface integrals and their applications
  • Proficiency in LaTeX for mathematical notation
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Homework Statement

Let's say I have a parametric surface [tex]\mathbf{r}(u,v) = <x(u,v),y(u,v),z(u,v)>[/tex]

Let's say I have to solve some surface integral

[tex]\iint f(x,y,z)\;dS[/tex]

Can I do this differential?

[tex]dS = |\mathbf{r_u} \times \mathbf{r_v}|dA[/tex]

also, has anyone noticed how small LaTeX has gotten on PF?
 
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flyingpig said:

Homework Statement




Let's say I have a parametric surface [tex]\mathbf{r}(u,v) = <x(u,v),y(u,v),z(u,v)>[/tex]

Let's say I have to solve some surface integral

[tex]\iint f(x,y,z)\;dS[/tex]

Can I do this differential?

[tex]dS = |\mathbf{r_u} \times \mathbf{r_v}|dA[/tex]

also, has anyone noticed how small LaTeX has gotten on PF?

What do you mean 'can I do it'? If the r_u and r_v are partial derivatives and dA means du*dv, then that's an expression for dS alright.
 
flyingpig said:
Let's say I have a parametric surface [tex]\mathbf{r}(u,v) = <x(u,v),y(u,v),z(u,v)>[/tex]

Let's say I have to solve some surface integral

[tex]\iint f(x,y,z)\;dS[/tex]

Can I do this differential?

[tex]dS = |\mathbf{r_u} \times \mathbf{r_v}|dA[/tex]


absolutely
 

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