Prerequisite Quickstart

This is a quickstart guide to the prerequisites for quantum field theory.

Table of Contents

• Table of Contents
• Calculus 1/2
  • Derivatives
  • Integrals
  • Series
• Calculus 3
  • Vectors
  • Scalar Functions of Multiple Variables
  • Vector Functions of Single Variable
  • Vector Functions of Multiple Variables
  • Laplacian
  • Theorems

Calculus 1/2

Derivatives

If represents the position of a particle at time , then the average velocity of the particle over the time interval is given by

As , the average velocity approaches the instantaneous velocity, which is defined as

This limit is called the derivative of with respect to , and is denoted in Leibniz notation as , in Newtonian notation as , and in Lagrange notation as .

We can always take second derivatives, which are defined as the derivative of the derivative. For example, the derivative of the velocity is called the acceleration, and is denoted as , , or .

For single-variable functions, the derivative can be visualized as the slope of the tangent line to the curve at a given point. If the slope is zero, then the function is at a local maximum or minimum. If the second derivative is positive, then the function is concave up, and if it is negative, then the function is concave down. If the second derivative is zero at a point, there is an inflection point at that point.

First derivative	Second derivative	Info
		Increasing, concave up
		Increasing, concave down
		Decreasing, concave up
		Decreasing, concave down
		Local minimum
		Local maximum
		Horizontal inflection point
		Non-horizontal inflection point

Here are some common derivatives:

and three useful rules:

1. Chain rule:

   For example:

2. Product rule:

   For example:

3. Quotient rule:

   For example:

Implicit differentiation is a technique used to find the derivative of a function that is defined implicitly, rather than explicitly. A full understanding of implicit differentiation requires some multivariable calculus, but the basic idea is to think of two variables and as functions of a third variable . For example, consider the equation , which describes a circle.

We make these variables functions of :

Then, we can differentiate both sides with respect to , using the chain rule:

Multiplying both sides by gives us

which we can rearrange to get

Integrals

Now suppose we know the velocity of a particle as a function of time, . To find out how far the particle has traveled over a time interval, we can add up the infinitesimal distances traveled over each infinitesimal time interval. Each infinitesimal distance is given by , and we sum them over the time interval :

As , this sum approaches something called the integral, which is written as:

This is called the definite integral of from to . The formal statement of what we derived is the Fundamental Theorem of Calculus, which states that if is a differentiable function, then

where is the upper limit of the integral. There are a few techniques for evaluating integrals, most of which are the reverse of the techniques for evaluating derivatives:

1. Substitution: This is the reverse of the chain rule. For example, if we have

   we can use the substitution , which means . Substituting gives us

2. Integration by parts: This is the reverse of the product rule. For example, if we have

   we can use the formula

   where and . This gives us

There are some others that essentially boil down to using algebraic manipulation to rewrite the integral in a more convenient form.

Series

A sequence is a list of numbers. A series is the sum of a sequence.

For example, the sequence is the sequence of natural numbers. The series is the sum of the natural numbers. The series diverges, meaning it does not converge to a finite value.

The partial sum of a series is the sum of the first terms of the series:

where is the th term of the series. The series converges if the limit of the partial sum exists as :

The series diverges if the limit does not exist.

Below are the tests for convergence and divergence of series:

1. Divergence test: If , then the series diverges.

2. Integral test: If is a positive, continuous, and decreasing function for , then the series converges if and only if the integral converges:

3. Comparison test: If for all , then the series converges if and only if the series converges.

4. Limit comparison test: If and for all , then the series converges if and only if the series converges:

   where is a positive constant.

5. Ratio test: If , then the series converges if and diverges if .

6. Root test: If , then the series converges if and diverges if .

7. Alternating series test: If is a decreasing sequence of positive numbers and , then the series converges.

We can approximate any function as a power series, known as the Taylor series expansion. It is defined so that all the derivatives of the function at a point are equal to the derivatives of the Taylor series at that point:

where is the th derivative of at . The Taylor series expansion is a powerful tool for approximating functions, especially when the function is difficult to evaluate directly. Here are some common Taylor series expansions around :

Calculus 3

Vectors

A vector can be thought of as a directed line segment. It has a magnitude (length) and a direction.

It is often represented as an arrow, with the tail of the arrow at the origin and the head of the arrow at the point in space. A vector can be constructed as a linear combination of basis vectors. Let be a vector with length 1 pointing in the direction, be a vector with length 1 pointing in the direction, and be a vector with length 1 pointing in the direction. Then, we can write any vector as

where , , and are the components of the vector.

The vector can also be represented as a column vector:

The length of the vector is given by the norm or magnitude of the vector, which is defined as

A unit vector is a vector with length 1. It is often denoted with a hat, like this: .

A linear transformation is a transformation that preserves the operations of vector addition and scalar multiplication. For example, the transformation defined by

is a linear transformation, because it preserves vector addition and scalar multiplication. More precisely, a linear transformation satisfies the following properties:

where and are vectors and is a scalar. It can be represented as a matrix, which is a rectangular array of numbers.

A matrix can be thought of as a linear transformation that takes a vector as input and produces another vector as output. For example, the matrix

acts on the vector as follows:

The matrix is a diagonal matrix, because all the non-diagonal elements are zero.

Scalar Functions of Multiple Variables

Consider a function of two variables. The function can be thought of as a surface in three-dimensional space. Such a function is called a scalar field.

We can also think of and as two components of a vector in two-dimensional space. Then the function can be written as .

When we take a derivative, we need to consider which variable we are taking the derivative with respect to. The partial derivative of with respect to is defined as the derivative of with respect to , while holding constant. This is denoted as , , , , or (there are many notations). Formally, it is defined as

The partial derivative of with respect to is defined similarly:

The gradient of is a vector whose direction is the direction of steepest ascent of the function . Its magnitude is the rate of change of the function in that direction. It is denoted as . In Cartesian coordinates, it has the convenient form

The directive of in the direction of a unit vector is the rate of change of the function in that direction. It is denoted as or . It can be calculated as

Next, consider two inputs and . The change in from to is given by the sum of contributions from the change in and the change in , similar to the chain rule:

This is known as the multivariable chain rule. We can write this equation with a vector as

This is similar to the single-variable chain rule, where we have . As such, we can think of the gradient as a generalization of the derivative to multiple variables.

The reason the gradient points in the direction of steepest ascent can be seen from the following argument. Suppose we have a function and we want to find the directional derivative in the direction of a unit vector . The value of that is , where is the angle between the gradient and the unit vector. The maximum value of this expression occurs when , or when the gradient and the unit vector are in the same direction. Thus, the directional derivative is maximized in the direction of the gradient.

Taking the multivariable chain rule in Equation , we can consider what happens with a non-infinitesimal change in and . In other words, suppose the input goes along a path from to . What is the change in along that path?

We can write the change in as the accumulation of the infinitesimal changes in along the path:

As the number of infinitesimal changes goes to infinity, we can replace the sum with an integral:

This is known as a line integral, and what we have derived is the fundamental theorem of line integrals or gradient theorem.

Vector Functions of Single Variable

A vector function is a function that takes a single variable as input and produces a vector as output. For example, the function is a vector function of the single variable . The function can be thought of as a curve in three-dimensional space.

The function can be differentiated with respect to to give the velocity of the particle at time :

Next, suppose we have a velocity vector and we want to find out how much the particle has moved over a time interval . We can do this by integrating the velocity vector over the time interval:

This is similar to the line integral we derived earlier, but in this case, we are integrating over a time interval instead of a path in space.

The curvature of a curve is a measure of how much the curve deviates from being a straight line. Its value is given by the formula

For a curve in two dimensions, the curvature is given by

Vector Functions of Multiple Variables

A vector function of multiple variables is a function that takes multiple variables as input and produces a vector as output. It is also called a vector field.

For example, the function is a vector function of three variables. As always, we can think of the vector function as a function of a single vector variable .

The flux of a vector field through a surface is defined as the integral of the vector field over the surface. The intuition comes from the idea of the vector field as a flow of fluid. Split the surface into pieces of area . The flux through each piece of area is given by the dot product of the vector field and the area vector , where is the unit normal vector to the surface.

The total flux through the surface is given by the integral of the flux over the surface:

The flux is a measure of how much of the vector field passes through the surface. A related concept is the divergence of a vector field, which is a measure of how much the vector field spreads out from a singular point. We can find it by considering the flux-per-unit-volume as the volume shrinks to zero:

The divergence is defined as the limit of the flux-per-unit-volume as the volume shrinks to zero:

As the notation suggests, in Cartesian coordinates, the divergence of a vector field is given by

The divergence is a scalar field, which means it takes a vector as input and produces a scalar as output.

The circulation of a vector field is defined as the line integral of the vector field around a closed curve :

The circulation is a measure of how much the vector field "circulates" around the closed curve. A related concept is the curl of a vector field, which is a measure of how much the vector field "curls" around a point. We can find it by considering the circulation-per-unit-area as the area shrinks to zero:

The curl is defined as the limit of the circulation-per-unit-area as the area shrinks to zero:

Unlike the divergence, the curl is a vector field, which means it takes a vector as input and produces a vector as output.

In Cartesian coordinates, the curl of a vector field is given by

Laplacian

The Laplacian of a scalar field is defined as the divergence of the gradient of the scalar field:

It is similar to the second derivative of a function of a single variable, which is defined as the derivative of the derivative.

It can be shown that the Laplacian is a measure of the average value of the function in a small neighborhood around a point. In other words, if we take a small volume around a point and average the value of the function over that volume, the Laplacian is the rate of change of that average value as the volume shrinks to zero. The Laplacian is a scalar field, which means it takes a vector as input and produces a scalar as output.

If the Laplacian of a function is zero, then the function is said to be harmonic.

Theorems

There are a few theorems that are useful for working with vector fields.

1. Divergence theorem/Gauss's theorem, applicable for a surface enclosing a volume :

2. Stokes' theorem, applicable for a curve bounding a surface :

3. Gradient theorem, applicable for two points enclosing a curve :

The following are some useful vector calculus identities: