🚨 Limited Offer: First 50 users get 500 credits for free — only ... spots left!
Free flashcards to ace your A-level - Further Mathematics
Learn faster with 46 A-level flashcards. One-click export to Notion.
Learn fast, memorize everything and ace your A-level. No credit card required.
Want to create flashcards from your own textbooks and notes?
Let AI create automatically flashcards from your own textbooks and notes. Upload your PDF, select the pages you want to memorize fast, and let AI do the rest. One-click export to Notion.
Further Mathematics
46 flashcards
Further Mathematics is an additional A-level qualification that expands on the content covered in A-level Mathematics. It includes topics like complex numbers, matrices, hyperbolic functions, and more advanced calculus.
Complex numbers are numbers that can be expressed in the form a + bj, where a and b are real numbers and j is the imaginary unit (√(-1)). They are used to solve equations that have no real number solutions.
A matrix is a rectangular array of numbers or algebraic expressions arranged in rows and columns. Matrices are used to represent systems of linear equations and perform linear transformations.
Hyperbolic functions are analogs of the trigonometric functions that are based on the hyperbolic angle. The main hyperbolic functions are sinh, cosh, tanh, coth, sech, and csch.
A differential equation is an equation that relates a function with its derivatives. It describes how a quantity changes in relation to another quantity.
Polar coordinates are a way of representing points in a plane using a distance (radius) from the origin and an angle from a fixed direction (polar axis).
A vector is a quantity that has both magnitude and direction. Vectors are commonly represented by arrows and are used to describe quantities like force, velocity, and displacement.
A group is a set of elements with a binary operation that satisfies four group axioms: closure, associativity, existence of an identity element, and existence of an inverse for every element.
Proof by induction is a method of mathematical proof used to establish a statement for all natural numbers. It involves proving the base case and then showing that if the statement holds for one case, it must also hold for the next case.
A Taylor series is an infinite series representation of a function around a given point. It is based on the idea of expressing a function as an infinite sum of its derivatives at that point.
A partial differential equation (PDE) is an equation that involves partial derivatives of an unknown function with respect to multiple independent variables.
A Fourier series is an expansion of a periodic function as a sum of sines and cosines of different frequencies. It is a way of representing a periodic signal as the sum of simple oscillating functions.
The fundamental theorem of calculus establishes the relationship between the two main operations of calculus: differentiation and integration. It states that differentiation and integration are inverse operations.
A vector field is an assignment of a vector to each point in a space. It describes a vector quantity that has both magnitude and direction across the entire space.
A line integral is an integral where the function to be integrated is evaluated over a curve. It is used to calculate the work done by a force over a path or the mass of a wire with variable density.
A differential form is a tensor quantity that can be integrated over regions of space. It generalizes the concept of a scalar field to allow tensors and vectors to be integrated over surfaces and volumes.
Green's theorems relate line integrals around simple closed curves to double integrals over the regions that the curves enclose. They provide a basis for transferring properties between line and double integrals.
A manifold is a topological space that locally resembles Euclidean space near each point. It is a generalization of the concept of a curve or surface to higher dimensions.
A Lie group is a group that is also an analytic manifold, with the property that the group operations are analytic functions. Lie groups are used in various areas of mathematics and physics.
Topology is the mathematical study of the properties of geometric objects that are preserved under continuous deformations such as stretching, twisting, or bending.
A field is a set of elements with two binary operations (addition and multiplication) that satisfy the field axioms, including commutativity, associativity, distributivity, and the existence of identity and inverse elements.
A ring is a set of elements with two binary operations (addition and multiplication) that satisfy the ring axioms, including commutativity and associativity of addition, and distributivity of multiplication over addition.
A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. It maps straight lines to straight lines.
A basis is a set of linearly independent vectors in a vector space that can be used to represent any vector in the space as a linear combination of the basis vectors.
A determinant is a scalar value that is a function of the entries of a square matrix. It is used to solve systems of linear equations and study the properties of linear transformations.
The Jordan canonical form is a matrix decomposition that puts a square matrix into a canonical form that is particularly useful for studying the behavior of the matrix under repeated multiplication.
The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric about its mean. It is widely used in statistics and probability theory.
A Markov chain is a stochastic process that satisfies the Markov property, which states that the future states depend only on the present state and not on the past states.
A Monte Carlo method is a computational algorithm that uses repeated random sampling to obtain numerical results. It is used to approximate solutions to complex problems that are difficult to solve analytically.
A stochastic process is a collection of random variables that represent the evolution of some system over time. It is used to model phenomena that involve uncertainty or randomness.
A Poisson process is a type of stochastic process that counts the number of events occurring within a given time interval, assuming that the events occur independently at a constant rate.
A differential operator is an operator that maps one function to another function by applying a combination of derivatives. Examples include the gradient, divergence, and Laplacian operators.
The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies. It is used in signal processing, image analysis, and various areas of physics.
A variational principle is a principle in physics that states that a certain quantity remains stationary (at a minimum, maximum, or saddle point) for the true solution of a physical system.
A conformal mapping is a transformation that preserves local angles between curves in a plane or on a surface. It is used in complex analysis, cartography, and other areas.
A fractal is a geometric shape that exhibits a repeating pattern at every scale. It has a fractional dimension and can be generated by a simple recursive process.
Chaos theory is the study of unpredictable and seemingly random behavior in deterministic nonlinear systems that are highly sensitive to initial conditions.
Game theory is the study of mathematical models of strategic interactions between rational decision-makers. It is used in economics, political science, and various other fields.
Cryptography is the practice and study of techniques for secure communication in the presence of third parties. It involves the use of codes and ciphers to protect information from unauthorized access.
Numerical analysis is the study of algorithms and methods for solving problems that are difficult or impossible to solve analytically. It involves approximation techniques and computational methods.
Mathematical modeling is the process of creating a mathematical representation of a real-world system or phenomenon to study, predict, and analyze its behavior.
Combinatorics is the branch of mathematics that deals with counting, arranging, and enumerating discrete structures, such as sets, permutations, and combinations.
Graph theory is the study of mathematical structures that model pairwise relations between objects. It has applications in computer science, biology, transportation networks, and many other fields.
Knot theory is the study of mathematical knots and their properties, such as knot invariants, knot polynomials, and knot groups. It has applications in physics, chemistry, and biology.
Algebraic geometry is the study of geometric objects defined by polynomial equations, using techniques from abstract algebra. It has applications in various areas of mathematics and physics.
Mathematical logic is the study of the formal systems used in mathematics, including their syntax, semantics, and deductive systems. It provides a foundation for many areas of mathematics.