Year 1 Sem 2/LC Mathematics for Physicists 1B/Lecture 01

Lecture 1: Partial Differentiation I: Intro and Welcome
Wed 21 Jan 2026 11:00

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Course Welcome

Recommended books:

• Mathematical Techniques 4e, Jordan & Smith
• Engineering Mathematics 8e, Stroud
• Calculus (Schaum), 6e, Ayres & Mendelson
• Advanced Calculus (Schaum), 6e, Ayres & Mendelson

Assessment details:

• Maths 1A/1B form a single 20 credit module.
• 80% assessed by a 3 hour exam - Section 1 is 36% with 6 short questions and Section 2 is 64% with 4 long questions.
• 20% assessed by problem sheets.

Course structure:

1. Partial Differentiation
   • Definition, total differential, chain rule, gradient.
   • Taylor series, stationary points, Lagrange multipliers.
2. Differential Equations
   • Definition, 1st order separable, exact and homogenous.
   • Linear equations: general solution, 1st order and constant coefficients.
3. Integration
   • Definition as area under the curve, fundamental theorem of calculus.
   • Integration by: substitution, parts, partial fractions and tricks.
4. Multiple Integrals
   • Multiple and repeated integrals. Change of order of integration.
   • Change of variables and the Jacobian. Arc length. Solids of revolution.

Multivariate Functions

Lots of physics involves functions of more than one variable. A physical quantity defined at every point in space is called a field. We can have both scalar fields and vector fields.

For example, some scalar fields are:

• $V(x, y, z)$: Electrostatic potential. This is often easier to work with compared to the full electric (vector) field.
• $T(x, y, z)$: Temperature.
• $p(x, y, z)$: Pressure.

While some vector fields are:

• $\underline{E}(x, y, z)$: Electric Field.
• $\underline{B}(x, y, z)$: Magnetic Field.
• $\underline{v}(x, y, z)$: Velocity Field (i.e in fluid mechanics).

Partial Derivatives

Consider a function of two variables. The partial derivative of a function with respect to one variable is the rate of change of a function wrt that variable, while keeping other variables constant. Effectively, we carry out a derivative while treating the other variables as if they were constants.

Suppose we have a function $f(x, y)$. The definition of a partial derivative is:

$$\frac{\partial f}{\partial x}(x_0, y_0) = \lim_{h \to 0} \frac{f((x_0 + h), y_0) - f(x_0, y_0)}{h}$$

$$\frac{\partial f}{\partial y}(x_0, y_0) = \lim_{k \to 0} \frac{f(x_0, (y_0 + k)) - f(x_0, y_0)}{k}$$

Just like we denote $\frac{df}{dx}$ as the derivative of a function of a single variable, we denote $\frac{\partial f}{\partial x}$ as the partial derivative of a function of several variables.

Note that this is not delta f and delta y, i.e. not $\frac{\delta f}{\delta x}$

In theory, we’d explicitly notate:

$$\left(\frac{\partial f}{\partial x}\right)_y$$

With the subscript $y$ explicitly stating that $y$ is being kept constant. This is rarely, but sometimes, needed. Consider $f(x, y, z) = x^2 \sin yz$. We have:

$$\frac{\partial f}{\partial x} = 2x \sin yz$$

$$\frac{\partial f}{\partial y} = x^2 z \cos yz$$

$$\frac{\partial f}{\partial z} = x^2 y \cos yz$$

Higher Orders

Higher derivatives are defined as they were previously, but they can now be mixed. For example, with $f(x, y) = x^2 \sin y$, we can write:

$$\frac{\partial f}{\partial x} = 2x \sin y \qquad \frac{\partial f}{\partial y} = x^2 \cos y$$

$$\frac{\partial^2 f}{\partial x^2} = 2 \sin y$$

We can also have:

$$\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial}{\partial y} \left(\frac{\partial f}{\partial x}\right) = 2x \cos y$$

Shorthand notation exists, i.e. $f_{xx}$ = $\frac{\partial^2 f}{\partial x^2}$ or $f_{yx} = \frac{\partial^2 f}{\partial y \partial x}$

For most cases, but not all, mixed derivatives are often independent of the order of partial derivatives, so: $f_{xy} = f_{yx}$

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