Exercises

Math practice problems

4956 practice problems, each with its own page, answer and explanation.

Practice sheets

Set 10

LDDMM 10

Arithmetic 10

Subset 10

Cartesian product 10

Function 11

Binary operation 10

Group 11

Identity element 10

Inverse element 11

Abelian group 10

Proposition 10

Logical operator 10

Negation 10

Conjunction 11

Disjunction 10

Implication 10

Logical equivalence 10

NAND 10

Truth table 11

Tautology 10

Contradiction 11

Predicate 10

Universal quantifier 10

Existential quantifier 10

Negation of quantifiers 10

Order of quantifiers 10

Axiomatic system 11

Rule of inference 10

Modus ponens 11

Formal proof 10

Consistency 10

Logical completeness 10

Gödel’s incompleteness 10

Limit of a function 20

One-sided limits 10

Limit laws 10

Indeterminate forms 10

Squeeze theorem 10

Special trigonometric limits 11

Limits at infinity 10

Asymptotes 10

Continuous function 20

Types of discontinuity 10

Intermediate value theorem 20

Extreme value theorem 20

Derivative 20

Tangent line 10

Basic differentiation rules 10

Product rule 10

Quotient rule 10

Chain rule 10

Derivatives of trigonometric functions 10

Derivatives of exponentials and logarithms 10

Implicit differentiation 10

Critical points 10

Local maxima and minima 10

Concavity and inflection points 10

Optimization problems 10

Related rates 10

L'Hôpital's rule 20

Rolle's theorem 20

Mean value theorem 20

Antiderivative 10

Indefinite integral 11

Riemann sums 10

Definite integral 10

Fundamental theorem of calculus 20

Integration by substitution 10

Integration by parts 10

Area between curves 10

Smooth function 10

Lie group 11

Smooth diffeomorphic deformation 11

Minimum energy 11

Diffeomorphic flow 11

Velocity field 11

Time derivative 11

System of linear equations 10

Matrix 10

Matrix multiplication 10

Gaussian elimination 10

Inverse matrix 10

Transpose matrix 10

Determinant 10

Cofactor and Laplace expansion 10

Properties of the determinant 10

Cramer's rule 10

Vector in ℝⁿ 10

Vector space 10

Vector subspace 10

Linear combination and span 10

Linear independence 10

Basis and dimension 10

Rank of a matrix 10

Linear transformation 10

Kernel and image 10

Matrix of a linear transformation 10

Rank–nullity theorem 10

Change of basis 10

Eigenvalue and eigenvector 10

Characteristic polynomial 10

Eigenspace 10

Diagonalization 10

Dot product 10

Vector norm 10

Orthogonal vectors 10

Orthogonal projection 10

Gram–Schmidt process 10

Sample space 10

Event 10

Probability measure 10

Classical probability 10

Addition rule 10

Permutations and combinations 10

Conditional probability 10

Independence of events 10

Law of total probability 10

Bayes' theorem 10

Random variable 10

Probability mass function 10

Probability density function 10

Cumulative distribution function 10

Expected value 10

Variance 10

Covariance 10

Bernoulli distribution 10

Binomial distribution 10

Poisson distribution 10

Uniform distribution 10

Exponential distribution 10

Normal distribution 10

Chebyshev inequality 10

Law of large numbers 10

Central limit theorem 10

Random sample 10

Sample mean 10

Estimator 10

Confidence interval 10

Hypothesis test 10

p-value 10

Differential equation 10

Order and linearity 10

Solution of a differential equation 10

Initial value problem 10

Existence and uniqueness theorem 10

Direction field 10

Separable equation 10

First-order linear equation 10

Integrating factor 10

Exact equation 10

First-order homogeneous equation 10

Bernoulli equation 10

Equilibrium points 10

Exponential growth and decay 10

Logistic model 10

Newton's law of cooling 10

Second-order linear equation 10

Superposition principle 10

Wronskian 10

Characteristic equation 10

Undetermined coefficients 10

Variation of parameters 10

Harmonic oscillator 10

Laplace transform 10

Laplace transform of a derivative 10

Inverse Laplace transform 10

Unit step function 10

Convolution 10

System of differential equations 10

Linear system and eigenvalues 10

Phase plane 10

Stability of an equilibrium 10

Cross product 10

Lines and planes in space 10

Parametric curve 10

Arc length 10

Quadric surface 10

Function of several variables 10

Level curve 10

Limit of a function of several variables 10

Partial derivative 10

Tangent plane 10

Differentiability in several variables 10

Gradient 10

Directional derivative 10

Chain rule in several variables 10

Critical point of a multivariable function 10

Hessian matrix 10

Second derivative test 10

Lagrange multipliers 10

Absolute extrema on a region 10

Double integral 10

Fubini's theorem 10

Double integral over a general region 10

Polar coordinates 10

Triple integral 10

Cylindrical and spherical coordinates 10

Change of variables in multiple integrals 10

Vector field in the plane and space 10

Line integral of a scalar function 10

Line integral of a vector field 10

Conservative field 10

Parametric surface 10

Surface integral 10

Divergence 10

Curl 10

Green's Theorem 10

Stokes' Theorem 10

Divergence Theorem 10

Direct proof 10

Proof by contradiction 10

Mathematical induction 10

Strong induction 10

Set operations 10

Inclusion-exclusion principle 10

Multiplication principle 10

Factorial 10

Permutations with repetition 10

Pigeonhole principle 10

Partial order relation 10

Hasse diagram 10

Graph 10

Binomial coefficient 10

Binomial theorem 10

Combinations with repetition 10

Degree of a vertex 10

Path in a graph 10

Connected graph 10

Tree 10

Eulerian cycle 10

Hamiltonian cycle 10

Bipartite graph 10

Graph coloring 10

Planar graph 10

Recurrence relation 10

Divisibility 10

Greatest common divisor 10

Euclidean algorithm 10

Modular congruence 10

Prime number 10

Homogeneous linear recurrence 10

Bézout's identity 10

Extended Euclidean algorithm 10

Euclid's lemma 10

Fundamental theorem of arithmetic 10

Least common multiple 10

Infinitude of primes 10

Linear Diophantine equation 10

Complete residue system 10

Modular inverse 10

Linear congruence 10

Chinese remainder theorem 10

Fermat's little theorem 10

Euler's totient function 10

Euler's theorem 10

Wilson's theorem 10

Multiplicative order 10

Primitive root 10

Quadratic residue 10

Legendre symbol 10

Euler's criterion 10

Law of quadratic reciprocity 10

Multiplicative arithmetic function 10

Divisor functions 10

Perfect number 10

Möbius function 10

Möbius inversion formula 10

Continued fraction 10

Convergent of a continued fraction 10

Pell's equation 10

Sum of two squares 10

RSA cryptosystem 10

Fermat primality test 10

Prime number theorem 10

Subgroup 10

Order of a group 10

Order of an element 10

Generated subgroup 10

Cyclic group 10

Center of a group 10

Group homomorphism 10

Group isomorphism 10

Kernel of a homomorphism 10

Coset 10

Index of a subgroup 10

Lagrange's theorem 10

Normal subgroup 10

Quotient group 10

First isomorphism theorem 10

Direct product of groups 10

Symmetric group 10

Cycle of a permutation 10

Sign of a permutation 10

Group action 10

Orbit and stabilizer 10

Orbit-stabilizer theorem 10

p-group 10

Sylow theorems 10

Classification of finite abelian groups 10

Dihedral group 10

Simple group 10

Unit of a ring 10

Zero divisor 10

Characteristic of a ring 10

First isomorphism theorem for rings 10

Polynomial ring 10

Polynomial division algorithm 10

Euclidean domain 10

Irreducible element 10

Unique factorization domain 10

Field of fractions 10

Multinomial coefficient 10

Exponential generating function 10

Integer composition 10

Integer partition 10

Partition generating function 10

Pentagonal number theorem 10

Stirling number of the second kind 10

Bell number 10

Stirling number of the first kind 10

Catalan number 10

Derangement 10

Burnside's lemma 10

Lattice or order theory 10

Steiner triple system 10

Vitali set 10

Measurable function 10

Minkowski inequality 10

Riesz–Fischer theorem 10

Hahn–Jordan decomposition 10

Convexity and the Hessian 10

Subgradient 10

Optimization problem 10

Local and global minima of convex functions 10

First-order necessary condition 10

Linear least squares 10

Line search 10

Wolfe conditions 10

Convergence with Lipschitz gradient 10

Newton's method for optimization 10

Proximal operator 10

Proximal gradient method 10

Bisection method 10

Fixed-point iteration 10

Secant method 10

Newton divided differences 10

Interpolation error 10

Runge phenomenon 10

Cubic splines 10

Cholesky factorization 10

Jacobi and Gauss-Seidel methods 10

Trapezoidal rule 10

Simpson's rule 10

Gaussian quadrature 10

Finite differences for derivatives 10

Euler's method 10

Local and global error 10

Runge-Kutta methods 10

Linear time-invariant systems 10

Impulse response 10

Continuous Fourier transform 10

Fourier transform properties 10

Aliasing 10

Discrete Fourier transform 10

Discrete transfer function 10

Poles, zeros and stability 10

Short-time Fourier transform 10

2D convolution on images 10

Ordered field 10

Supremum and infimum 10

Completeness axiom 10

Archimedean property 10

Absolute value 10

Density of the rationals 10

Convergent sequence 10

Uniqueness of limits 10

Monotone sequence 10

Monotone convergence theorem 10

Subsequence 10

Bolzano-Weierstrass theorem 10

Cauchy sequence 10

Sequential continuity 10

Uniform continuity 10

Differentiability and continuity 10

Partition and Darboux sums 10

Riemann integral 10

Integrability of continuous functions 10

Numerical series 10

Comparison test 10

Absolute convergence 10

Uniform convergence 10

Complex number 10

Complex conjugate 10

Modulus of a complex number 10

Polar form 10

De Moivre's formula 10

Complex nth root 10

Topology of the complex plane 10

Complex function 10

Complex limit 10

Complex continuity 10

Complex exponential 10

Complex logarithm 10

Complex trigonometric functions 10

Complex derivative 10

Cauchy-Riemann equations 10

Holomorphic function 10

Harmonic function 10

Möbius transformation 10

Curve and contour 10

Contour integral 10

Cauchy-Goursat theorem 10

Complex primitive 10

Cauchy integral formula 10

Liouville's theorem 10

Fundamental theorem of algebra 10

Complex Taylor series 10

Laurent series 10

Isolated singularity 10

Zero and pole 10

Residue 10

Residue theorem 10

Real integral by residues 10

Metric space 10

Open ball 10

Open set in a metric space 10

Convergence in a metric space 10

Complete metric space 10

Topology 10

Topological space 10

Discrete and trivial topology 10

Closed set 10

Basis of a topology 10

Neighborhood of a point 10

Interior of a set 10

Closure of a set 10

Boundary of a set 10

Accumulation point 10

Continuous function between topological spaces 10

Homeomorphism 10

Subspace topology 10

Product topology 10

Quotient topology 10

Hausdorff space 10

Connected space 10

Continuous image of a connected space 10

Path-connected space 10

Connected component 10

Open cover 10

Compact space 10

Compact sets in Hausdorff spaces 10

Continuous image of a compact space 10

Heine-Borel theorem 10

Sequential compactness 10

Lebesgue number 10

Set membership 10

Union of families 10

Intersection of families 10

Relative complement 10

Power set 10

Binary relation 10

Equivalence relation 10

Equivalence class 10

Injective function 10

Surjective function 10

Bijective function 10

Image and preimage 10

Finite set 10

Infinite set 10

Equipotence 10

Countable set 10

Cantor's diagonal theorem 10

Cardinal number 10

Schröder–Bernstein theorem 10

Well-ordering 10

Transfinite induction 10

Axiom of choice 10

Zorn's lemma 10

Zermelo–Fraenkel axioms 10

Construction of the naturals 10

Construction of the reals 10

Composition and inverse 30