Fundamental Theorems of (Vector) Calculus

Scott N. Walck

September 10, 2021 Fundamental Theorem of Calculus

The integral of the derivative of a function is equal to the difference of the values of the function at the upper and lower limits. Let f : R → R be a function.

Z b df dx = f (b) − f (a) a dx Gradient Theorem

The dotted line integral of the gradient of a scalar field over a curve is equal to the difference of the values of the scalar field at the end points of the curve.

C ~r1

~r2 Let C be a curve from ~r1 to ~r2. Let f be a scalar field (a scalar function of space), so that f (~r1) is a scalar. Then ∇~ f is a vector field. Z ∇~ f · d~` = f (~r2) − f (~r1) C Stokes’ Theorem

The flux integral of the curl of a vector field over a surface is equal to the dotted line integral of the vector field over the curve describing the boundary of the surface. C S Let S be a surface with boundary curve C. Let F~ be a vector field. Z Z (∇~ × F~) · d~a = F~ · d~` S C Divergence Theorem

The volume integral of the divergence of a vector field is equal to the flux integral of the vector field over the boundary surface.

V S Let V be a volume with boundary surface S. Let F~ be a vector field. Z Z (∇~ · F~)dτ = F~ · d~a V S Notation

Let I VF be the set of all vector fields, I SF be the set of all scalar fields, I Vol be the set of all volumes, I Srf be the set of all surfaces, I Crv be the set of all curves, I Pts be the set of all pairs of points in space, and I R be the set of all real numbers. Boundary

I The boundary of a volume is a surface. If V is a volume, then ∂V is a (closed) surface. I The boundary of a surface is a curve. If S is a surface, then ∂S is a (closed) curve. I The boundary of a curve is a pair of points. If C is a curve, then ∂C is a pair of points. Gradient Theorem

grad × id SF × Crv VF × Crv

id × ∂ R

SF × Pts eval, subtract R

grad × id (f , C) (∇~ f , C)

id × ∂ R

R (f , (~r1, r~2)) f (~r2) − f (~r1) = (∇~ f ) · d~` eval, subtract C Stokes’ Theorem

curl × id VF × Srf VF × Srf

id × ∂ R

VF × Crv R R

curl × id (F~, S) (∇~ × F~, S)

id × ∂ R

~ R F~ · d~` = R (∇~ × F~) · d~a (F , ∂S) R ∂S S Divergence Theorem

div × id VF × Vol SF × Vol

id × ∂ R

VF × Srf R R

div × id (F~, V ) (∇~ · F~, V )

id × ∂ R

~ R F~ · d~a = R (∇~ · F~)dτ (F , ∂V ) R ∂V V