2  Calculus

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

  1. Differential Equations Section 2.1, Vector Calculus Section 2.2
  2. Variational Calculus I, Variational Calculus II

2.1 Differential Equations

Introduction to ODEs and PDEs, see also Variational Calculus.


expanded notes: notes

test questions: quiz

2.1.1 Key results Small o and Big O notation

  • \(f(x_0 + h) = f(x_0) + f'(x_0)h + o(h)\)
  • \(f(x)=f(x_0) + (x-x_0)f'(x_0) + \frac{(x-x_0)^2}{2}f''(x_0) + ... \frac{(x-x_0)^n}{n!} f^{(n)}(x_0) + O((x-x_0)^{n+1})\) 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 \(f=c\). 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 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. 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 \(O(\delta t^2)=o(\delta t)\). 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


See this detailed explanation of differential forms by Terry Tao

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.

Example Integrate 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 \(y=-\frac{1}{2}x+1\). 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 \(\frac{\partial (x,y)}{\partial (u,v)}\)

Example Polar 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.