# 2 Calculus

Multivariate calculus. Very applied, so flirting with physics in this approach. For example intuition might be based on e.g. electrostatics (gulp!)

- Differential Equations Section 2.1, Vector Calculus Section 2.2
- Variational Calculus I, Variational Calculus II

## 2.1 Differential Equations

Introduction to ODEs and PDEs, see also Variational Calculus.

Progress:

expanded notes: notes

test questions: quiz

### 2.1.1 Key results

#### 2.1.1.1 Small o and Big O notation

- \(f(x_0 + h) = f(x_0) + f'(x_0)h + o(h)\)
- \(f(x)=f(x_0) + (x-x_0)f'(x_0) + \frac{(x-x_0)^2}{2}f''(x_0) + ... \frac{(x-x_0)^n}{n!} f^{(n)}(x_0) + O((x-x_0)^{n+1})\)

#### 2.1.1.2 Solution strategies for 1st order ODEs

- Direct Integration
- Separation of variables
- Integrating Factor - convert to (uv)‘=u’v+uv’
- Exact Given \(Q(x,y)\frac{dy}{dx} + P(x,y)=0\) is exact if there is a function \(f\) such that \(df\) is equal to \(dx\) LHS (also \(df=0\)), then we have \(\frac{\partial f}{{\partial x}}=P\) and \(\frac{\partial f}{\partial y}=Q\). If the domain is
*simply-connected*then this is true if \(\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}\).

Can solve by integrating \(\frac{\partial f}{\partial x}=P\) where the integrating constant is \(h(y)\), then differentiate w.r.t. \(y\) and use \(Q\) to find \(h'(y)\). Final solution is then \(f=c\).

#### 2.1.1.3 Bestiary of Differential Equations

*First Order*No second derivative, similarly*Nth Order**Homogenous*\(y=0\) is a solution, i.e. there is no*forcing term**Linear*x does not appear…

### 2.1.2 Glossary

*Liebniz’ Rule* Repeated application of the product rule for simple derivatives, i.e. \[ f^{(n)}(x)=\sum_{r+0}^n \binom{n}{r}u^{(r)}v^{(n-r)} \]

*Phase Space* For a n-th order diff eq., the n-dimensional space given by \(y\), \(y'\), \(\dots y^{(n-1)}\). An initial value e.g. \((y, y')\) is a vector in this space. The solutions are linearly independent and trace a surface in phase space.

*Wronskian* The determinant of \(\begin{vmatrix}y_1 &y_2 \\ y'_1 & y'_2 \end{vmatrix}\) is non-zero iff the solutions are independent, i.e. if the vectors are independent in phase space, then the solutions of the diff eq are independent. (Abel’s theorem): the Wronskian is either zero for all \(x\) or non-zero for all \(x\).

## 2.2 Vector Calculus

Introduction vector calculus. Progress:

expanded notes: notes

test questions: quiz

### 2.2.1 Curves, Scalar Fields and Vector Fields

#### 2.2.1.1 Maps taxonomy

A *parameterized curve* is a map \(\mathbb{R} \rightarrow \mathbb{R^n}\) a parameter (e.g. t, heuristically time) runs over the domain, and the image of the function \(\textbf{x}(t)\) is a line in the codomain. The image of \(\textbf{x}(s,t)\) from \(\mathbb{R}^2 \rightarrow \mathbb{R}^3\) would be a surface - for a fixed \(t\) corresponding to a point \(\textbf{x}\), there is one degree of freedom, \(s\). Note the choice of parameter does not affect the image.

A scalar field is a map \(\mathbb{R}^n \rightarrow \mathbb{R}\) associating a scalar value with each point in a vector space, e.g. temperature, Higgs field in the standard model of particle physics.

A vector field is a map \(\mathbb{R}^n \rightarrow \mathbb{R}^m\) associating a vector value with each point in a vector space, e.g. electric field and magnetic field.

#### 2.2.1.2 Curves

Can define the derivative \(\dot{\textbf{x}}(t)\) by \(\delta \textbf{x}(t)= \textbf{x}(t+\delta t) - \textbf{x}(t) = \dot{\textbf{x}}(t)\delta t + o(\delta t)\) or casually substitute \(O(\delta t^2)=o(\delta t)\).

#### 2.2.1.3 Line Integrals

** Scalar field**, say \(\phi(\textbf{x}(t))\).

Don’t want the line integral to depend on parameterisation, so can define using the arc length, e.g. from \(\textbf{a}=s_a\) to \(\textbf{b}=s_b\) we have:

\[ \int_C ds \, \phi = \int_{s_a}^{s_b} ds \, \phi(\textbf{x}(s))=\int_{t_a}^{t_b} dt \, \phi(\textbf{x}(t)) \frac{ds}{dt} = \int_{t_a}^{t_b} dt \, \phi(\textbf{x}(t)) |\phi(\dot{\textbf{x}}(t))| \]

So always positive and the \(\dot{\textbf{x}}\) cancels the Jacobian on a change of variables so the line integral does not depend on parameterization.

** Vector field**, say \(\textbf{F}:\mathbb{R}^n \rightarrow \mathbb{R}^n\).

So how to integrate \(\textbf{F}\) along some curve \(C\). Useful to get a single number for a line integral, so we integrate the component of the vector field tangent to the curve, i.e. to integrate from \(\textbf{x}(t_a)=\textbf{a}\) to \(\textbf{x}(t_b)=\textbf{b}\):

\[ \int_C \textbf{F}(\textbf{x})\cdot d\textbf{x}=\int_{t_a}^{t_b}dt \, \textbf{F}(\textbf{x}(t)) \cdot \dot{\textbf{x}}(t) \]

So this can be positive or negative, an *orientation* given by the tangent vector \(\dot{\textbf{x}}(t)\).

Generally obviously the line integral depends on both the thing you’re integrating \(\textbf{F}\) and the choice of curve \(C\).

Can decompose a curve \(C=C_1+C_2+\dots\) into piecewise smooth curves joined at the endpoints, and the line integral is the sum of the line integrals of each piece. If \(-C\) is the curve \(C\) with the opposite orientation, then \(\int_{-C}=-\int_C\). If we have a closed curve, then can write the line integral as the *circulation of* \(\textbf{F}\) *around* \(C\):

\[ \oint_C \textbf{F}(\textbf{x})\cdot d\textbf{x} \]

If this is always zero, this implies the integral between any two points does not depend on the curve \(C\), physicists call this a *conservative field* 🤷♂️

A vector field \(\textbf{F}\) is conservative iff \(\textbf{F}=\nabla \phi\) for some scalar field \(\phi\).

Note that \(\nabla \phi\) is *always* a vector, e.g. in \(\mathbb{R}^3\) for a scalar field \(\phi(x,y,z) \, \mathbb{R}^3 \rightarrow \mathbb{R}\) the gradient can be packaged as a vector field \(\nabla \phi = \frac{\partial \phi}{\partial x}\hat{\textbf{x}}+\frac{\partial \phi}{\partial y}\hat{\textbf{y}}+\frac{\partial \phi}{\partial z}\hat{\textbf{z}}\) for the usual orthonormal basis \(\textbf{e}_i\).

Can check a field is conservative by the property \(\partial_i F_j= \partial_j F_i\) since both are \(\frac{\partial^2}{\partial x^i \partial y^i} \phi\). Line integral of a conservative vector field is the analog of the integral of a total derivative, and depends just on the endpoints.

### 2.2.2 Differential Forms

Given a function \(\phi(\textbf{x})\) on \(\mathbb{R}^n\) The *differential* is a function of \(\textbf{x}\) defined as: \[d\phi = \frac{\partial \phi}{\partial x^i} dx^i=\nabla \phi \cdot d\textbf{x}\] where e.g. \(d\textbf{x}=\Bigl(\begin{smallmatrix}dx\\dy\\dz\end{smallmatrix}\Bigr)\)

Taking the inner product of a vector field \(\textbf{F}(\textbf{x})\) on \(\mathbb{R}^n\) with the infinitesimal vector above gives the *differential one-form*, \(\textbf{F}(\textbf{x})\cdot d\textbf{x}\) the one bit indicating we should think of integrating over a curve. (General case of the differential above, if \(\textbf{F}(\textbf{x})\cdot d\mathbb{x}=d\phi\) the differential form is said to be *exact*).

### 2.2.3 Surfaces and Volumes

For an infinitesimal area \(dA\) can define a Riemannian integral (note limits) \[ \int_D \phi(x,y)dA=\int_a^b \,dy \int_{x_1(y)}^{x_2(y)}\,dx \phi(x.y)\]

** Fubini’s theorem** (proven 1907) states that for suitably well-behaved functions and regions, all ways of decomposing the integral agree.

ExampleIntegrate the function \(\phi(x,y)=x^2y\) over the triangle formed from {(0,0),(0,1),(2,0)}. Note this region \(D\), is bounded by \(y=-\frac{1}{2}x+1\).

#### 2.2.3.1 Jacobian

Changing coordinates from \((x,y)\) to \((u,v)\) we have \(\Bigl(\begin{smallmatrix}\delta x\\ \delta y \end{smallmatrix}\Bigr)=\Bigl(\begin{smallmatrix}\frac{\partial x}{\partial u} \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u} \frac{\partial y}{\partial v}\end{smallmatrix}\Bigr)\Bigl(\begin{smallmatrix}\delta u \\ \delta v\end{smallmatrix}\Bigr)\)

Notationally can write the Jacobian as \(\frac{\partial (x,y)}{\partial (u,v)}\)

ExamplePolar coordinates \(x=r\cos\theta\) and \(y=r\sin\theta\) \[\frac{\partial (x,y)}{\partial (r,\theta)}=\begin{vmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{vmatrix} = r\]

### 2.2.4 Glossary

**Arc length** \(s=\int_{t_0}^t d\lambda \, |\dot{\textbf{x}}(\lambda)|\) since \(\delta s = |\delta \textbf{x}|+O(\delta t^2) =\dot{\textbf{x}}(t)\delta t + O(\delta t^2)\)

**Tangent vector** to a parameterized curve \(\textbf{x}(t)\) is \(\dot{\textbf{x}}(t)\) arc length does not depend on parameter, if curve is parameterized by arc length then tangent vector \(\textbf{t}\) has unit magnitude, since \(\frac{ds}{dt}=|\dot{\textbf{x}}|\)

**Curvature** \(\kappa \textbf{n}\) is the derivative of the unit tangent vector, \(\ddot{\textbf{x}}(s)\) intuitively captures how different the curve is from a straight line (which has zero curvature).

**Osculating Plane** defined by the principal normal \(\textbf{n}\) and the tangent vector \(\textbf{t}\). To a parametrized curve.