AMath 501
Outline
Basic Vector Manipulations
- Dot products and cross products (area of a parallelogram)
- Relation between norms and dot products
- Scalar triple products (signed volume)
- Geometric understanding of the concepts above
Coordinates & Position Vectors
- Cartesian Coordinates:
- Spherical Coordinates:
- Cylindrical Coordinates:
Curvilinear Coordinates
- Unit vectors:
- Scale factors:
- All scale factors in coordinates are stated above.
Derivatives and Arc Length
- Derivative of position vector
:
- This can be written as:
- Arc length differential
:
- In orthogonal coordinates.
- Arc Length Formula:
Gradient Operator
for a scalar ,
- Geometric explanation: Represents the steepest ascent.
Divergence Operator
- In Cartesian coordinates:
- In general:
where:
- Write down div in spherical/cylindrical coordinates.
- Physical explanation: Represents the net flux out of a volume per unit volume.
Curl Operator
Physical Explanation
- The circulation around a small area per unit area.
Line Integral
- Parameterizing the line integral:
Conservative Field
-
If
, where is a scalar. -
If
is conservative, then:
- The independence of the path:
Ifis conservative, then .
The opposite way is also true.
Surface Integral
- We have to find
and . - In general:
The surfacecan be expressed in terms of and .
- If the surface is written as
, then:
Surface Representation
- If we use
to write the surface as , then:
- Since:
and:
- If
, then:
Surface Area
Divergence Theorem
- Compute the net flux through the surface of a volume by using the volume integral:
- Where:
Stokes' Formula/Theorem
- The circulation around an area
, where is the boundary of . - RHS (Right-Hand Side) is easier to compute.
This expands to:
In 2D:
Definition of Holomorphic Functions / Analytic Functions
A holomorphic function (or analytic function) is a complex function
exists and is the same no matter how
Cauchy-Riemann Relations:
For
Branch Cuts / Points:
Some functions (like
: Has a branch cut along the negative real axis. : Often has a branch cut along the negative real axis to ensure single-valuedness.
Example
- Holomorphic Function:
Here,
and
- Non-Holomorphic Function:
Here,
Cauchy Theorem
If
Key Idea:
The integral of an analytic function over a closed loop is zero if the function is analytic everywhere inside and on the contour.
Example:
Let
since
Cauchy Integral Formula
If
Key Idea:
The value of a holomorphic function inside a contour can be computed using the values of the function on the contour.
Example:
Let
Since
Residue
Intuition: Circulation Around the Singularity
The residue measures how much the function “circulates” around the singularity. If you imagine walking along the contour
In
Key Points About Residues and Singularities
- Residue Explains Local Behavior Near a Singularity
The residue at a singularitydescribes the “circulation” of near that singularity.
For example, in:
and are the two singularities. - Each singularity has its own residue, calculated by isolating the behavior of
near that singularity (locally).
- Residues Are Independent of Distance From the Singularity
Residues depend only on the singularity itself, not on the distance from it. The contourused to calculate the residue can have any radius, as long as it encloses only that singularity (and no others).
This is because the integral
- Residue Measures Local Circulation
Residue tells us how“circulates” locally around the singularity. This is like isolating the behavior of at that point.
- Each residue at
and is independent of the other singularity.
- Why Distance Doesn’t Affect Residue
The residue at a singularity depends only on the coefficient ofin the Laurent series expansion. Since Laurent series expansions focus on the singularity and ignore the behavior elsewhere, the residue is unaffected by the distance from the singularity.
Residue Theorem
Let
Key Idea:
The integral around a closed contour is determined by the sum of residues of the function’s singularities inside the contour.
Single Singular Point at z = 0
In your specific example, the contour
The residue of