∫0(this is retarded) 1 dx

It is currently 17:42, and I am in the basement of the library. I have a test that I was going to take Friday.. er... Saturday... uh.. today... I guess tomorrow. I am sincerely dead. Let me tell you about the grand lewdities that are encompassed in my math test here.

1. Line Integrals.

• W=∫F·dr= ∫F(r(t))·r'(t)
• If F is a force field, this integral represents work performed in traversing the curve.
• Parametrization for r when broken up amongst multiple curves is given through (1-t)r₀+(t)r₁

2. Other coordinate systems

• Cylindrical Coordinates: x=r·cosθ y=r·sinθ x²+y²=r² and dV=drdθdz
  
• Spherical Coordinates: x=(ρ)cos(θ)sin(φ) y=(ρ)sin(θ)sin(φ) z=(ρ)cos(φ) and dV=dρdφdθ

3. Divergence, Gradient, Curl

• Divergence div(F)=∂M/∂x + ∂N/∂y +∂R/∂z = ∇·F
• Curl curl(F)=(∂R/∂y-∂N/∂z)i-(∂R/∂x-∂M/∂z)j+(∂N/∂x-∂M/∂y)k = ∇ x F

4. Green's Theorem

• ∫F·dr=∫∫_R(∂N/∂x-∂M/∂y)dA
• Area of R=1/2 ∫(-y)dx+x dy

5. Surface Integrals

• ∫∫_R(1+(f_x)²+(f_y)²)^1/2 dA=surface area of.. well... a surface
• ∫∫_R(φ(x,y,f(x,y))(1+(f_x)²+(f_y)²)^1/2 dA=integration of a function over a given surface

6. Stokes's Theorem

• ∫F·dr=∫∫_Scurl(F)·n dS
• I sincerely have no idea what this is used for...

7. The Divergence Theorem

• ∫∫_SF·n dS=∫∫∫_Udiv(F)dV
• Again, not a clue.

Let it be noted that there were supposed to be a few"C" subscripts after various integral signs, and also that some of the integral signs are supposed to have little circles in them, but 1. I was too lazy to go back and insert the subscripts after I figured out how to make them and 2. I didn't know how to make the fancier integral sign.

I am going to try to study some more now.