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Mathematica Lesson 14: Surface Integrals

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​When asked "Do you want to automatically evaluate all the initialization cells in the notebook...?" I recommend "No" so that you can work through each cell one-at-a-time.

Parametrizations

Question 2

You can visualize a parametric surface in Mathematica using ParametricPlot3D. For example, the paraboloid over the disk of radius 4 in the xy-plane is
In[]:=
ParametricPlot3D[{u*Cos[v],u*Sin[v],u^2},{u,0,4},{v,0,2*Pi}]
Below, fill in the command that would visualize the part of the plane x+y+z=10 inside the cylinder x²+y²=9:
ParametricPlot3D[{u*Cos[v],Answer,Answer},{u,0,Answer},{v,0,2*Pi}]

Question 3

To create a parametric function in Mathematica, we can do​
​r[u_,v_]:={ the parametrization... }​
​Fill in the code below to parametrize the part of the plane 2x+y+z=6 in the first octant.
r[u_,v_]:={u,v,Answer}
(We don't include the domain when we write the parametrization in Mathematica. The domain is used in the bounds of integration.)

Line Integrals

Question 4

Given a parametrization r⃗ (u,v), the differential of surface area is dS=‖ r⃗_u × r⃗_v ‖dA, where the subscripts denote partial derivatives and dA stands for either dudv or dvdu.​
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​Fill in the Mathematica code below that would compute dS for the saddle surface z=xy:
(* set up the parametrization and integrand components*)​​r[u_,v_]:={u,v,Answer}​​ru=D[r[u,v],u]​​rv=D[r[u,v],Answer]​​n=Cross[Answer]
(* compute ‖ r⃗_u × r⃗_v ‖ *)​​dS=Answer[n.n]//FullSimplify
In the Mathematica codes below, we will use dS as shorthand to represent the quantity ‖ r⃗_u × r⃗_v ‖.

Question 5

To compute the surface area of a parametric surface r(u,v), (u,v)∈D (the domain for the parameters u and v), we evaluate ∬S 1 dS= ∬D ‖ r⃗_u × r⃗_v ‖dA.​
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​Fill in the Mathematica code below that would compute the surface area of the sphere x²+y²+z²=4 in spherical coordinates.
(* set up the parametrization and integrand components*)​​r[u_,v_]:={2*Cos[u]Sin[v],Answer,Answer}​​ru=D[r[u,v],u]​​rv=D[r[u,v],Answer]​​n=Cross[Answer]
(* compute ‖ r⃗_u × r⃗_v ‖ *)​​dS=Answer[n.n]//FullSimplify
(* now integrate! *)​​Integrate[Integrate[Answer,{u,Answer,Answer}],{v,Answer,Answer}]
Does this agree with our familiar formula 4πR² (from Archimedes)?

Question 7

To compute a scalar surface integral ∬S f dS, we follow these steps from lecture:
◼
  • Define the parametrization r⃗(u,v) and specificy the domain D for (u,v).
    • ◼
  • Compute f(r⃗(u,v)).
    • ◼
  • Compute r⃗_u(u,v), r⃗_v(u,v), and compute the length of ‖ r⃗_u × r⃗_v ‖.
    • ◼
  • Setup and solve ∬S f dS = ∬D f(r(u,v))‖ r⃗_u × r⃗_v ‖ dA, where dA represents either dudv or dvdu.
    • Fill in the code to compute this in Mathematica for the function f(x,y,z)=x²y²z over the portion of the cone z=sqrt(x²+y²) that lies between the planes z=1 and z=5:
    (* Set up the parametrization and the function *)​​r[u_,v_]:={u*Cos[v],u*Sin[v],Answer}​​f[{x_,y_,z_}]:=x^2*y^2*z
    (* Compute r⃗_u(u,v), r⃗_v(u,v), and compute the length of ‖ r⃗_u × r⃗_v ‖. *)​​ru=Answer​​rv=Answer​​n=Cross[Answer]//FullSimplify​​dS=Answer[n.n]//FullSimplify​​​​(* Set up and integrate ∬D f(r(u,v))‖ r⃗_u × r⃗_v ‖ dA *)​​Integrate[Integrate[f[r[u,v]]dS,{u,Answer,Answer}],{v,Answer,Answer}]
    Compute the answer to the nearest hundredth for Moodle.

    Question 8

    To compute a vecor surface integral (flux integral) ∬S F⃗ ⋅ dS⃗ , we follow these steps from lecture:
    ◼
  • Define the parametrization r⃗(u,v) and specificy the domain D for (u,v).
    • ◼
  • Compute F⃗ (r⃗(u,v)).
    • ◼
  • Compute r⃗_u(u,v), r⃗_v(u,v), and compute r⃗_u × r⃗_v . Check if this is correctly oriented relative to the surface. (If it isn't, put a minus sign in front of your computation in Step 4.)
    • ◼
  • Setup and solve ∬S F⃗ ⋅ dS⃗  = ∬D F(r(u,v)) ⋅ ( r⃗_u × r⃗_v ) dA, where dA represents either dudv or dvdu.
    • Fill in the code to compute the flux of the vector field F⃗(x,y,z)=⟨ y, x, x+z² ⟩ across the paraboloid z=x²+y² over the unit disk, oriented with upward pointing normal vector.
    (* Set up the parametrization and the function *)​​r[u_,v_]:={u*Cos[v],u*Sin[v],Answer}​​F[{x_,y_,z_}]:={y,x,x+z^2}
    (* Compute r⃗_u(u,v), r⃗_v(u,v), and r⃗_u × r⃗_v. *)​​ru=Answer​​rv=Answer​​n=Cross[Answer]//FullSimplify
    (* Set up and integrate ∬D F(r(u,v)) ⋅  r⃗_u × r⃗_v  dA *)​​Integrate[Integrate[F[r[u,v]].Answer,{u,Answer,Answer}],{v,Answer,Answer}]
    Compute the exact answer for Moodle.