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Summary for Vector Calculus and Complex Calculus
(Math 321)
By Lei Li
1 Vector Calculus
1.1 Parametrization
Curves, surfaces, or volumes can be parametrized. Below, I’ll talk about 3 D case. Suppose we use ~ex, ~ey, ~ez as the basis and then the position vector for any point on the curve (surface, volume) can be written as:
~r = x~ex + y~ey + z~ez
To parametrize, we need to write x, y, z as functions of our parameters. You need to choose the parameters wisely so that our problem would be easy to solve. Something you must know:
- For curve, the degree of freedom is one and thus you should only have ONE parameter in your parametrization
- You should have 2 independent parameters for surface
- 3 independent parameters for volume Example:
- For the circle (x − a)^2 + (y − b)^2 = r^2 , we can use the angle and x = a + r cos θ, y = b + r sin θ and then your parametrization would be ~r(θ) = (a + r cos θ)~ex + (b + r sin θ)~ey
- For the function graph y = f (x), x itself is a convenient parameter.
- for the region 0 ≤ x, y ≤ a, z = b, x, y would be good parameters. Two important parametrization would be to use the spherical coordinates and cylindrical coordinates. Hope you know the mean- ings the parameters. To get the Cartesian equation from the parametriza- tion, you just need to eliminate the parameters. Example: x = a tan θ, y = b sec θ. What is ~r? What is the Cartesian equation?
1.2 Curves and line integrals
Suppose you get the parametrization already (given or obtained by y- ourself) ~r(t):
d~r =
d~r dt
dt = ~r′(t)dt
Sometimes, we are given by functions like ~r(t) = ~r(θ(t)) or ~r(t) = ~r(u(t), v(t)). It’s easy to get the following by chain rule:
~r′(t) =
d~r dθ
θ′(t)
~r′(t) =
∂~r ∂u
u′(t) +
∂~r ∂v
v′(t)
Having obtained d~r, everything would be just easy:
∫ (^) t 2
t 1
|r~′(t)|dt
You should know how to parametrize the curve using arclength param- eter.
- Area enclosed by the initial vector, end vector and the curve:
A =
|~r × d~r|
F^ ~ · d~r
Example: Along the curve y = sin x, 0 ≤ x ≤ π, if the force is F~ = F 0 ~r′. Calculate the work done by this force if the particle moves from the origin to (π, 0). Example: #6 in this section
1.3 surface and surface integrals
We have the parametrization ~r(u, v) = x(u, v)~ex + y(u, v)~ey + z(u, v)~ez. Example: For the sphere: ~r(θ, φ) = r sin θ cos φ~ex + r sin θ sin φ~ey + r cos θ~ez For the torus ~r =...
1.4.1 Coordinate curves, coordinate surfaces and coordinate vec- tors
Suppose we have ~r(u, v, w) = x(u, v, w)~ex + y(u, v, w)~ey + z(u, v, w)~ez. What are the coordinate curves and coordinate surfaces? Calculate the co- ordinate vectors.
1.4.2 Line elements, Surface elements
Given ~r(u, v, w). What are the line elements for the coordinate curves? What is the general line element? What are the surface elements for the coordinate surfaces?
1.4.3 Volume element
dV = (~ru × ~rv) · ~rwdudvdw
In the three cases below find suitable ~r and then calculate the total mass inside the volume: The density of a kind of material is 2(H −h)^2. h is the height from the lowest point of that volume and H is the largest height.
- The ball x^2 + y^2 + z^2 ≤ 1
- The cylinder x^2 + y^2 = 1, 0 ≤ z ≤ 1
- The cube 0 ≤ x, y, z ≤ 1
- The solid in the first octant and bounded by x = 1, y = 1 and z = x^2 y^2
1.4.4 Orthogonal coordinates and scale factors
We can always write the coordinate vectors as its magnitude times the direction.
hi = |∂~r/∂qi|, qˆi =
hi
∂~r ∂qi
If the parameters are orthogonal, then ˆqi · qˆj = δij. If they satisfy right- handed rule furthermore, then we get a new basis. The surface element and
volume element in such cases are quite easy:
dS~ 3 = qˆ 3 h 1 h 2 dq 1 dq 2 dV = h 1 h 2 h 3 dq 1 dq 2 dq 3
Example: Calculate ~er, ~eθ, ~eϕ for spherical coordinates. (They are important when we derive the gradient, divergence, etc under spherical coordinates)
1.5 Change of variables for integration
Just remember one thing: Jacobian! Example: Calculate the Jacobian if we change from 2D Cartesian coor- dinates to polar coordinates. Use this to evaluate
−∞ e
−x^2 dx
1.6 ∇ operator, gradient, curl, divergence
1.6.1 In Cartesian coordinate
- f (x, y, z) is a given nice scalar function. ∇f is called the gradient of function f , which points to the direction of fastest increasing and its magnitude equals the rate of change with respect to the distance going from one point. According to this meaning:
df = d~r · ∇f
Your task: Using the Cartesian form of d~r =... to get:
∇f =
∂f ∂x
~ex +
∂f ∂y
~ey +
∂f ∂z
~ez
According to its geometric meaning and this expression, it’s quite nat- ural to get the direction derivative:
∂f ∂n
= ˆn · ∇f
- It’s convenient to define a vector operator now:
∇ = ~ex
∂x
∂y
∂z
You just assume that ∇f = u~er + v~eθ + w~eϕ because it’s a vector and then you’ll get:
∂f ∂r
dr +
∂f ∂θ
dθ +
∂f ∂ϕ
dϕ = |~rr|udr + |~rθ|vdθ + |~rϕ|wdϕ
Then you’ll have
u =
∂f /∂r hr
∂f ∂r
v =
∂f /∂θ hθ
r
∂f ∂θ
w =
∂f /∂ϕ hϕ
r sin θ
∂f ∂ϕ
which gives the following important formula:
∇ = ~er
∂r
r
∂θ
r sin θ
∂ϕ
Using the definition of divergence, gradient, curl, Laplacian etc, one can get the expressions for them under spherical coordinates. I won’t list them here (You can find them online.). However, you should understand how to derive them even though you don’t have to memorize them. Example: In ∇ · ~v, we would have the term (~eθ (^1) r∂θ∂ ) · (u~er) which is nonzero. Please calculate this term out.
Other coordinates
Please derive ∇ operator using cylindrical coordinates and compare your results with those on Wikipedia.
1.7 Fundamental theorems for vector calculus
C
(F dx + Gdy) =
A
∂G
∂x
∂F
∂y
)dxdy
LHS can be also written as
C ~v^ ·^ d~r^ if we define^ ~v^ =^ F~ex^ +^ G~ey
One can check that the above formula is nothing but a special case of Stokes’ theorem if the surface is restricted in the x-y plane. The curl form of Green’s Theorem: ∮
C
~v · d~r =
A
∇ × ~v · ~ez dA
is just exactly the same formula as above. Furthermore, if we define ~n as the unit normal vector with the suitable direction, we can write the left hand side as
C (G~ex^ −^ F~ey)^ ·^ ~ndr, and thus we have
C (G~ex^ −^ F~ey)^ ·^ ~ndr^ =^
A(
∂G ∂x −^
∂F ∂y )dxdy.^ Change the notations and we can have ∮
C
(F~ex + G~ey) · ~ndr =
A
∂F
∂x
∂G
∂y
)dxdy
C
~v · ~ndr =
A
∇ · ~vdA
This is an analogy of Gauss’ Theorem in 2D and it’s called the diver- gence form of Green’s Theorem
S
∇ × ~v · dS~ =
C
~v · d~r
Here S is any surface in space with boundary C and C is a closed curve in space. Example: If S is a closed surface, C would be empty. What would the right hand side be? Use Gauss’ Theorem to justify this.
S
~v · dS~ =
S
~v · ~ndS =
V
∇ · ~vdV
Here, S is a closed surface (no boundary) and V is the volume enclosed by it (this means the boundary of V is C but C has no boundary).
Example: Show that
C ∇ ×^ f^ ·^ d~r^ = 0 if^ C^ is closed in space and^ f^ is a good function.
2.3 Cauchy-Riemann equations
We can regard any function of x, y as a function of z, f (z). We say it’s analytical if f ′(z) exists in a domain we are interested in. We know it’s analytical if it is the sum of a power series. However, we have a more fundamental criterion–Cauchy-Riemann equations. A function f (z) = u(x, y) + iv(x, y) is analytical if and only if
∂u ∂x
∂v ∂y ∂u ∂y
∂v ∂x
One amazing fact is that both u and v should be harmonic! There are problems in the last homework.
2.4 Complex Integrals
Understand the definition. You should know how to use parametriza- tion to calculate some simple complex integrals.∫ Example: How to prove
|z|=
1 z dz^ = 2πi? How to calculate^
|z|=2 z
∗dz?
2.4.1 Cauchy’s Theorem
If a function f (z) is analytical inside a closed curve C and on C( Actually, we only need it to be continuous on C), we then have: ∮
C
f (z)dz = 0
One important corollary of this theorem is ∮
C
(z − a)ndz =
2 πi n = − 1 and C encloses a 0 otherwise
2.4.2 Cauchy’s formula
If f (z) is analytical inside and on C, then: ∮
C
f (z) (z − a)n^
dz =
2 πif (n)(a) n!
if a is inside C and 0 otherwise. This means that we can determine everything about f (z) inside C only using values on C if f (z) is analytical. This is not quite amazing since the real and imaginary parts are harmonic.
2.4.3 Applications
Use Cauchy’s formula and Cauchy’s theorem to calculate some real in- tegrals. Example:
−∞
1 x^4 +1 dx^
0 cos(x
(^2) )dx etc.
