- #1
Rasalhague
- 1,387
- 2
I'm new to Mathematica. I used it to integrate the scalar field
[tex]f:\mathbb{R}^3 \to \mathbb{R} \; \bigg| \;f(x,y,x)=z^2[/tex]
over the top half of a unit sphere centered on the origin, paramaterising this surface with
[tex]\phi: \mathbb{R}^2 \rightarrow \mathbb{R}^3 \; \bigg| \; \phi(r,\theta)=(r \cos \theta, r \sin \theta, \sqrt{1-r^2})[/tex]
so that
[tex]f(\phi(r,\theta))=1-r^2.[/tex]
I set up the integral like this:
[tex]\int_R f(\phi(r,\theta))\left \| \partial_r \phi \times \partial_\theta \phi \right \| dr d\theta = \int_0^{2\pi} \int_0^1 r \sqrt{1-r^2} \; dr d\theta = \frac{2 \pi}{3}[/tex]
where the partial sign (curly d) with subscript variable stands for the partial derivative with respect to that variable, ||v|| denotes the norm (a.k.a. magnitude) of a vector v, and the times symbol, x, is the cross product of vectors.
In Mathematica, I was able to calculate this as follows:
phi={r*Cos[theta],r*Sin[theta],Sqrt[1-r^2]}; Integrate[phi[[3]]^2*Norm[Cross[D[phi,r],D[phi,theta]]],{theta,0,2*Pi},{r,0,1}]
and also, in the following two different ways, by plugging in the already simplified integrand:
Integrate[r*Sqrt[1-r^2],{theta,0,2*Pi},{r,0,1}]
Integrate[Integrate[r*Sqrt[1 - r^2], {r, 0, 1}], {theta, 0, 2*Pi}]
But the last of these methods didn't work when I used the unsimplified expression phi[[3]]^2*Norm[Cross[D[phi,r],D[phi,theta]]] in place of r*Sqrt[1 - r^2].
Integrate[
Integrate[
phi[[3]]^2*Norm[Cross[D[phi, r], D[phi, theta]]], {r, 0,
1}], {theta, 0, 2*Pi}]
It took a long time to calculate, then produced many lines of complicated symbolic expressions involving complex numbers and hyperbolic trig functions. A similar thing happened when I asked it to calculate just the inner integral:
Integrate[phi[[3]]^2*Norm[Cross[D[phi, r], D[phi, theta]]], {r, 0, 1}]
Can anyone tell me what went wrong: why the simplified expression worked with both methods, Integrate[ ,{ },{ }] and Integrate[Integrate[ ,{ }],{ }], but the equivalent full one only worked by the first method, Integrate[ ,{ },{ }]? Something to do with the order of operations that leads it to try dividing by something unpleasant, or am I making an elementary syntactic mistake?
[tex]f:\mathbb{R}^3 \to \mathbb{R} \; \bigg| \;f(x,y,x)=z^2[/tex]
over the top half of a unit sphere centered on the origin, paramaterising this surface with
[tex]\phi: \mathbb{R}^2 \rightarrow \mathbb{R}^3 \; \bigg| \; \phi(r,\theta)=(r \cos \theta, r \sin \theta, \sqrt{1-r^2})[/tex]
so that
[tex]f(\phi(r,\theta))=1-r^2.[/tex]
I set up the integral like this:
[tex]\int_R f(\phi(r,\theta))\left \| \partial_r \phi \times \partial_\theta \phi \right \| dr d\theta = \int_0^{2\pi} \int_0^1 r \sqrt{1-r^2} \; dr d\theta = \frac{2 \pi}{3}[/tex]
where the partial sign (curly d) with subscript variable stands for the partial derivative with respect to that variable, ||v|| denotes the norm (a.k.a. magnitude) of a vector v, and the times symbol, x, is the cross product of vectors.
In Mathematica, I was able to calculate this as follows:
phi={r*Cos[theta],r*Sin[theta],Sqrt[1-r^2]}; Integrate[phi[[3]]^2*Norm[Cross[D[phi,r],D[phi,theta]]],{theta,0,2*Pi},{r,0,1}]
and also, in the following two different ways, by plugging in the already simplified integrand:
Integrate[r*Sqrt[1-r^2],{theta,0,2*Pi},{r,0,1}]
Integrate[Integrate[r*Sqrt[1 - r^2], {r, 0, 1}], {theta, 0, 2*Pi}]
But the last of these methods didn't work when I used the unsimplified expression phi[[3]]^2*Norm[Cross[D[phi,r],D[phi,theta]]] in place of r*Sqrt[1 - r^2].
Integrate[
Integrate[
phi[[3]]^2*Norm[Cross[D[phi, r], D[phi, theta]]], {r, 0,
1}], {theta, 0, 2*Pi}]
It took a long time to calculate, then produced many lines of complicated symbolic expressions involving complex numbers and hyperbolic trig functions. A similar thing happened when I asked it to calculate just the inner integral:
Integrate[phi[[3]]^2*Norm[Cross[D[phi, r], D[phi, theta]]], {r, 0, 1}]
Can anyone tell me what went wrong: why the simplified expression worked with both methods, Integrate[ ,{ },{ }] and Integrate[Integrate[ ,{ }],{ }], but the equivalent full one only worked by the first method, Integrate[ ,{ },{ }]? Something to do with the order of operations that leads it to try dividing by something unpleasant, or am I making an elementary syntactic mistake?