PHYS2150 Methods in Physics I
The Complete Exam Bible · PHYS2150 Overview
- The University of Hong Kong
- First Semester, 2026-27
- Level 2 physics
- Two-hour written examination
- Analytical and numerical methods
What this course is built to teach
Methods in Physics I develops mathematical and computational problem solving for university physics.
- Assessment focus A written examination, tests and assignments reward complete mathematical setup and verification.
- Hardest step Choosing the correct geometric region, equation class or boundary case before calculating.
- Past material Earlier papers reveal question types and common errors, not the current paper format.
- How to prepare Rebuild solutions without notes, then verify units, conditions and limiting behaviour.
How PHYS2150 is assessed
| Component | Weight | Format |
|---|---|---|
| Written examination | 50% | Two-hour written examination |
| Tests | 40% | Tests scheduled within the teaching plan |
| Assignments | 10% | Six assignments including computational assignments |
The current course outline publishes these three components and their numeric weights total 100%. It states that the examination is written and lasts two hours. Confirm the current hurdle position, question count, sections and permitted materials on Moodle.
Assessment structure
Segment widths reproduce the three published weights, which total one hundred per cent.
Current PHYS2150 dates
| Date | Item | Control |
|---|---|---|
| 19 October 2026 | Test 1 | Listed in the published teaching plan. |
| 16 November 2026 | Test 2 | Listed in the published teaching plan. |
Dates are as published in the current published course outline. Confirm exact deadlines and submission settings in the live LMS.
What PHYS2150 covers
The guide follows the published teaching sequence from coordinate geometry and multivariable calculus through vector methods, differential equations and scientific computing.
Three-Dimensional Coordinates and Surfaces
lines, planes, distances, quadric surfaces and coordinate changesGeometry before algebra
Three-dimensional coordinate geometry becomes systematic once every object is represented by the information that defines it. A point needs three coordinates. A line needs one point and a direction vector.
- Vector and parametric line equations
- Plane normals and incidence
- Distances between geometric objects
- Quadric surfaces from traces
- Cylindrical and spherical coordinates
- Intersection and substitution checks
Partial Derivatives and Local Approximation
limits, differentiability, chain rules, tangent planes and Taylor approximationWhat the chapter connects
Limits establish whether a function settles to one value, differentiability supplies the local plane, the gradient selects a direction, and Taylor expansion extends that plane with curvature.
- Multivariable limits and paths
- Partial derivatives and gradients
- Differentiability and tangent planes
- Multivariable chain rules
- Taylor approximation and remainder
- Directional derivatives and level sets
Extrema, Constraints and Error Estimation
critical points, boundary checks, Lagrange multipliers and differential error boundsOne workflow, three decisions
This chapter separates candidate generation, classification and comparison. Derivative equations generate locations, curvature tests classify local behaviour, and domain-wide value checks establish global extrema.
- Critical points and Hessian tests
- Closed-region boundary checks
- Lagrange multiplier equations
- Differentials and sensitivity
- Absolute and relative uncertainty
- Constraint regularity and sensitivity
Double and Triple Integrals
regions, order reversal, polar, cylindrical and spherical integrationRegions come before antiderivatives
A double or triple integral accumulates a quantity over a region, so the first task is geometric. Draw the boundary curves or surfaces, locate their intersections, and choose a representative slice.
- Iterated integrals and slices
- Order reversal
- Polar area integrals
- Cylindrical and spherical volumes
- Jacobians and substitutions
- Mass and centre of mass
Vector Motion, Arc Length and Curvature
velocity, acceleration, Frenet vectors, arc length and osculating circlesA vector function carries a path and a clock
A space curve may be represented by r(t)=f(t)i+g(t)j+h(t)k. Its component domains must all be valid, and increasing t chooses an orientation.
- Vector functions and derivatives
- Velocity, speed and acceleration
- Arc length parametrisation
- Tangent and normal vectors
- Curvature and osculating circles
- Tangential and normal acceleration
Gradient, Divergence and Curl
scalar and vector fields, potentials and curvilinear coordinate operatorsBegin by identifying the field type
A scalar field assigns one number to every point, such as temperature, pressure, density or potential. A vector field assigns a vector, such as velocity, gravitational acceleration or electric field.
- Scalar and vector fields
- Gradient and level surfaces
- Divergence and sources
- Curl and rotation
- Conservative fields and potentials
- Laplacian and del identities
First-Order Differential Equations
linear, separable, homogeneous and exact equations with initial dataClassification controls the solution path
A first-order differential equation links an unknown function with its first derivative, but similar-looking equations can require different methods.
- Equation classification
- Separable equations and equilibria
- Linear integrating factors
- Homogeneous substitutions
- Exact equations and potentials
Linear Higher-Order Differential Equations
characteristic roots, Wronskians, forcing, resonance and particular solutionsOrder determines how many solutions are needed
A linear nth-order ordinary differential equation has a homogeneous solution space with n independent directions on an interval where the normalized coefficients are continuous.
- Characteristic equations
- Repeated and complex roots
- Wronskians and independence
- Undetermined coefficients
- Variation of parameters and resonance
Wave, Heat and Laplace Equations
boundary conditions, separation constants, eigenmodes and series solutionsThree equations encode different physics
Partial differential equations describe fields with several independent variables. The one-dimensional wave equation models propagating or vibrating displacement and contains a second time derivative.
- PDE types and physical models
- Dirichlet and Neumann conditions
- Separation of variables
- Eigenvalues and eigenmodes
- Initial data and modal series
Scientific Computing and Numerical Methods
graphics, fitting, root finding, minimisation and MATLAB programmingComputation belongs inside a validation loop
Scientific computing in this course connects plotting, curve fitting, root finding, minimisation and MATLAB programming with analytical physics.
- Plots with labelled physical axes
- Curve fitting and residuals
- Root finding and convergence tests
- Numerical minimisation
- Scripts, functions and debugging
The published outline explicitly contrasts that goal with a cookbook approach: students are expected to decide how to represent a problem, choose a method, carry it through analytically or numerically, and test whether the result makes physical sense.
The teaching sequence begins with three-dimensional coordinate geometry, continues through multivariable and vector calculus, and then moves into ordinary and partial differential equations. A scientific-computing strand adds plotting, curve fitting, root finding, minimisation and programming.
How the topics connect
Coordinate geometry supplies directions, normals, surfaces and coordinate transformations.
Multivariable calculus turns those objects into local rates, tangent planes, constrained extrema and integrals over physical regions. Vector functions describe motion, while gradient, divergence and curl turn fields into statements about rise, source strength and rotation.
Differential equations then model change: first-order equations introduce classification and initial conditions, higher-order equations add characteristic roots and forcing, and partial differential equations combine spatial boundaries with time evolution.
Computation closes the loop by making models testable through plots, residuals and convergence checks.
What assessment preparation should look like
The official assessment split is a written examination worth half the course, tests worth four tenths, and assignments worth the remaining tenth. The current outline describes the examination as written and two hours long.
Confirm its question count, sections and permitted aids in the current Moodle announcements. Earlier course papers and performance comments remain useful for recognising sustained multi-part mathematical problems and recurring losses: incomplete regions, missing unit directions, unfinished potentials, overlooked resonance modifications and discarded separation cases.
This guide uses original questions to rehearse those reasoning moves without treating an earlier paper as a prediction of the current format.
Worked example · free
Choose coordinates before integrating
- 2Use spherical coordinates because both boundaries are coordinate surfaces.
- 4State the complete radial and angular limits with the Jacobian.
- 2Check coverage and dimensional scaling.
Key terms
- Gradient
- A vector of partial derivatives pointing in the direction of greatest local increase.
- Divergence
- A scalar measuring the local net outward tendency of a vector field.
- Curl
- A vector measuring local rotational tendency and its axis.
- Jacobian
- A determinant that measures how a coordinate transformation scales area or volume.
- Curvature
- The rate at which a curve changes tangent direction per unit arc length.
- Wronskian
- A determinant used to test linear independence of candidate solutions.
- Integrating Factor
- A multiplier that converts a linear differential equation into a product derivative.
- Exact Equation
- A differential equation whose coefficients are partial derivatives of one potential function.
- Dirichlet Condition
- A boundary condition prescribing the value of the unknown field.
- Neumann Condition
- A boundary condition prescribing the normal derivative of the unknown field.
- Eigenmode
- A spatial pattern permitted by an operator and its boundary conditions.
- Residual
- The difference remaining when an approximate result is substituted into the original relation.
- Level Surface
- A surface on which a scalar field has one constant value.
- Osculating Circle
- The circle matching a curve's tangent and curvature at a point.
PHYS2150 FAQ
How is the course assessed?
The current outline assigns half of the mark to one written examination, four tenths to tests and the remaining tenth to assignments, including computational assignments. Those published weights total the whole course mark. No component-specific hurdle is stated in the captured outline, so consult Moodle for later operational announcements rather than inferring one.
What is known about the written examination?
The current outline states that it is a two-hour written examination worth half of the course mark. It does not publish a current number of questions, section structure or permitted aids. Earlier papers on the course site show useful mathematical question types, but their format belongs to an earlier offering and should not be treated as a promise.
Are past papers available?
The course site includes earlier Test 1 and Test 2 questions with solutions, plus an earlier examination and performance comments. Use them to identify methods, multi-part structure and common losses. Every practice problem in this guide is newly authored, and the older files do not establish the current examination layout.
Why are multiple integrals difficult?
The antiderivative is often not the hard part. The marks depend on describing the complete region, choosing an order or coordinate system, including the Jacobian and splitting the solid when one set of limits cannot cover it. A sketch and a coverage check should appear before evaluation.
How should vector-calculus answers be checked?
First confirm whether the requested output is a scalar, vector or potential. Then inspect units and geometry: gradients are normal to level surfaces, divergence describes source strength, and curl describes rotation. For a potential, differentiate the final scalar function and recover every field component.
What causes mistakes in forced differential equations?
A trial particular solution can overlap the complementary solution. When the forcing family duplicates a characteristic root, multiply the whole trial by enough powers of the independent variable to restore independence. Course performance comments specifically identify this modification rule as a source of lost credit.
How do boundary conditions change a PDE solution?
Boundary conditions select the allowed spatial modes. Fixed-value conditions often produce sine modes, while zero-normal-derivative conditions produce cosine modes and can retain a constant mode. Initial data then determine the amount of each permitted mode, so applying initial information before solving the boundary problem reverses the logical order.
What is the best revision routine?
For each topic, write the representation and method name from memory, complete one original problem, then verify the result by substitution, units or limiting behaviour. Keep an error log organised by branch decisions, such as a missed boundary, Jacobian, equilibrium solution, resonance multiplier or zero mode, because those decisions transfer across many questions.
Questions students ask in each PHYS2150 chapter
- When is a line parallel to a plane?
- Its direction vector is perpendicular to the plane normal, so their dot product is zero. That condition alone does not distinguish a separate parallel line from one lying in the plane. Substitute one line point into the plane equation before deciding whether a nonzero distance exists. Chapter 01 · Three-Dimensional Coordinates and Surfaces
- Do existing partial derivatives prove differentiability?
- No. Partial derivatives inspect coordinate directions, while differentiability requires one linear approximation to control simultaneous changes in all directions. Continuous first partial derivatives nearby give a standard sufficient condition, but exceptional points need direct analysis of the total increment and its remainder. Chapter 02 · Partial Derivatives and Local Approximation
- Why is solving the gradient equation insufficient?
- A zero gradient addresses only differentiable interior points and generates candidates rather than conclusions. A closed region can attain global extrema on smooth boundaries, corners, endpoints or singular points, so those locations must be analysed separately and their objective values compared. Chapter 03 · Extrema, Constraints and Error Estimation
- Why might a solid need more than one integral?
- The upper, lower or radial boundary can change across the projection. Splitting at the intersection curve faithfully describes the geometry, while forcing one set of limits can omit or double-count part of the solid. In spherical coordinates, compare competing radial endpoints and solve where they are equal; that angle is the natural place to divide the integral. Chapter 04 · Double and Triple Integrals
- What is the difference between velocity and speed?
- Velocity is a tangent vector carrying direction and rate, while speed is its nonnegative magnitude. A particle can move at constant speed and still accelerate because its velocity direction changes. Uniform circular motion is the standard example: speed is constant, tangential acceleration is zero, but inward normal acceleration has magnitude v² divided by the radius. Chapter 05 · Vector Motion, Arc Length and Curvature
- Does zero divergence mean a field is zero?
- No. It means local outward and inward flux balance. A circulating field can have zero divergence while remaining nonzero everywhere and may also have nonzero curl. For example, F=(-y,x,0) has zero divergence but curl (0,0,2). Always compute or interpret curl separately; the two operators measure independent local features of a field. Chapter 06 · Gradient, Divergence and Curl
- Why record equilibria before separating variables?
- Division by a factor involving the unknown function excludes values where that factor is zero. Those values may satisfy the original equation as constant solutions and often represent steady physical states. Record them first, then solve the nonconstant branches. A phase-line sign test also reveals whether each retained equilibrium attracts or repels nearby solutions. Chapter 07 · First-Order Differential Equations
- What does a repeated characteristic root change?
- A real root with multiplicity m contributes m functions: e^{rx}, xe^{rx}, and successive powers through x^{m-1}e^{rx}. The powers create the missing independent directions. Without them, the complementary solution has fewer arbitrary constants than the equation’s order and cannot generally satisfy the required initial data. Chapter 08 · Linear Higher-Order Differential Equations
- What is the difference between initial and boundary data?
- Boundary data act at the spatial edges throughout the time interval and determine which eigenfunctions are admissible. Initial data describe the field at the starting time and determine the coefficient of each permitted mode. A wave equation normally needs both initial displacement and initial velocity because it is second order in time. Chapter 09 · Wave, Heat and Laplace Equations
- What makes a scientific plot useful?
- Axes need variable names and units, and the displayed domain must expose the behaviour being tested. Sampling should resolve expected curvature or oscillation without joining across discontinuities. Distinguish data from model curves, use a meaningful scale, and write a caption that explains which visible pattern supports or challenges the analytical expectation. Chapter 10 · Scientific Computing and Numerical Methods
How to study for the exam
Organise revision around decisions rather than a formula list. For coordinate problems, practise extracting points, directions and normals. For calculus, decide whether the task is local approximation, optimisation or accumulation over a region. For differential equations, name the equation class and conditions before manipulating it.
After every complete solution, perform one independent check: substitute into the original equation, differentiate a recovered potential, verify boundary data, inspect units or compare a limiting case. Use earlier course materials to sample question types and marking emphasis, but time your rehearsal against the current two-hour duration only after Moodle confirms the operational format.
Keep computational exercises connected to the analytical chapters by plotting regions, fields or solution families and reporting a residual or convergence receipt.
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