MATH 213 MORE THOROUGH CHEAT SHEET

Question 1

List the steps for scalar surface integral that involve parametrization

∫ ∫_sf(x,y,z)dS

Answer 1

1. Parametrize the surface: r(u,v)
2. Find f(r(u,v))
3. dS=|r_ux r_v|dudv
4. Evaluate the integral

Question 2

List the steps for scalar surface integral that doesn’t involve parametrization.

∫ ∫_sf(x,y,z)dS

Answer 2

1. Write the surface as z=g(x,y)
2. Find f(x,y,g(x,y))
3. dS=(Check Below)
4. Evaluate Integral

Question 3

List the steps for vector functions along surface Integrals that need parametrization.

∫ ∫_sF*ndS= ∫ ∫_sF*dS=Flux

Answer 3

1. Parametrize surface: r(u,v)
2. Find F(r(u,v))
3. dS=(r_ux r_v)dudv
4. Evaluate the Integral

Question 4

List the steps for vector functions along surface integrals that do not need parametrization.

∫ ∫_sF*ndS= ∫ ∫_sF*dS=Flux

Answer 4

1. Write surface as z=g(x,y)
2. Find F(x,y,g(x,y))
3. dS=<-dg/dx,-dg/dy,1>*dA (up)
4. Evaluate Integral

Question 5

Stokes Theorem: Explain it please.

Answer 5

If S is a surface with boundary curve C, then ∳_cF*dr=∫ ∫(curlF)*dS, where curlF= ∇XF.

Question 6

Now Explain the Divergence Theorem Please!!!

Answer 6

If S is closed, then ∫ ∫F*dS= ∫ ∫ ∫_EdivF dV where E is the solid region contained in S and divF= ∇*F. S: is a positive outward orientation.

Question 7

Use Green’s Theorem Circ/Curl Form and explain what it is .

Answer 7

If C is a closed curve, then ∳_C F*Tds=∳Pdx+Qdy=∫ ∫_R(dQ/dx-dP/dy)dA where R is the two dimensional region bounded by C

Question 8

Explain the Fundamental Theorem for Line Integrals

Answer 8

If F is conservative [dQ/dx=dP/dy or curlF=0] then ∫_c F*dr=f(end)-f(beg) where f is the potential function for F.

Question 9

Then explain the steps for vector Functions with line integrals or bhasically work

∫_CF*Tds=∫_CF*dr=∫_CPdx+Qdy=Work

Answer 9

1. Parametrize curve: r(t)
2. find F(r(t))
3. dr=r’(t)dt
4. Evaluate integral

Question 10

Then for scalar functions ∫_cf(x,y,z)ds explain the steps

Answer 10

1. Parametrize curve: r(t)
2. Find f(r(t))
3. ds=|r’(t)|dt=√((dx/dt)²+(dy/dt)²+(dz/dt)²⁾dt
4. Evaluate Integral