Multivariable calculus

133 problems · Functions of several real variables.

Search problems Level

Ex.34Summary of continuity

#summary #continuity

Continuity

Ex.77Continuity of a rational function

#rational function #domain #continuity

Continuity

High school

Ex.107Coordinate projections are continuous

#projection #continuity #coordinates

Continuity

Ex.225Continuity of a polynomial

#continuity #polynomial #continuity everywhere

Continuity

High school

Ex.256Double integral over a rectangle

#double integral #rectangle #iterated

Multiple integrals

High school

Ex.257Evaluating partial derivatives at a point

#partial derivatives #evaluation #point

Partial derivatives

High school

Ex.258First partial derivatives

#partial derivatives #definition #computation

Partial derivatives

High school

Ex.379Double integral over a triangular domain

#double integral #iterated integration #triangular domain

Multiple integrals

Ex.380Partial derivatives and the gradient

#partial derivatives #gradient #chain rule

Partial derivatives

Ex.435Critical points of a polynomial

#critical points #polynomial #Hessian

Optimisation in several variables

Ex.436Summary of constrained optimization

#summary #constrained optimization

Optimisation in several variables

High school

Ex.517Separable integrand on a square

#separability #product of integrals #square

Multiple integrals

High school

Ex.518Summary of multiple integrals

#summary #multiple integrals

Multiple integrals

Ex.519Double integral over a triangular domain

#triangle #double integral #bounds

Multiple integrals

High school

Ex.520Double integral under a parabola

#double integral #type-I domain #parabola

Multiple integrals

High school

Ex.521A symmetric triangle integral

#symmetry #triangle #double integral

Multiple integrals

High school

Ex.522Directional derivative

#directional derivative #gradient #direction

Partial derivatives

High school

Ex.523Euler's theorem for homogeneous functions

#homogeneous functions #Euler #partial derivatives

Partial derivatives

Ex.524The gradient

#gradient #vector #direction

Partial derivatives

High school

Ex.591The divergence

#divergence #vector field #computation

Vector calculus

Ex.592Closed paths in conservative fields

#closed path #conservative #zero integral

Vector calculus

Ex.593Line integral of a scalar field

#line integral #scalar field #parametrization

Vector calculus

Ex.733Lagrange multipliers on a circle

#Lagrange #multipliers #equality constraint

Optimisation in several variables

Ex.734Optimisation under a linear constraint

#Lagrange #linear constraint #minimum

Optimisation in several variables

Ex.769Double integral in polar coordinates

#polar coordinates #double integral #Jacobian

Multiple integrals

Ex.770Triple integral over a simplex

#triple integral #iterated integration #simplex

Multiple integrals

Ex.771The chain rule for a composition

#chain rule #composition #directional derivative

Partial derivatives

Ex.794Line integral of a differential form

#line integral #differential form #parametrization

Vector calculus

Ex.857Boundary classification

#boundary #classification #circle

Optimisation in several variables

Ex.858Distance from point to plane

#Lagrange #distance #plane

Optimisation in several variables

Ex.859Two critical points and their nature

#critical points #saddle #maximum

Optimisation in several variables

Ex.860Lagrange multipliers on a circle

#Lagrange #circle #extrema

Optimisation in several variables

Ex.861Optimization under a linear constraint

#Lagrange #linear constraint #minimum

Optimisation in several variables

Ex.862Maximizing a product

#Lagrange #product #constraint

Optimisation in several variables

Ex.863A unique critical point that is a saddle

#saddle #unique critical #Hessian

Optimisation in several variables

Ex.864A discontinuous function with bounded values

#discontinuity #bounded #counterexample

Continuity

Ex.865Discontinuity via two paths

#discontinuity #paths #limits

Continuity

Ex.866Norm and distance are continuous

#norm #distance #continuity

Continuity

Ex.867Continuity of a piecewise function

#piecewise #continuity #xy

Continuity

Ex.868Continuity via polar coordinates

#polar #continuity #bound

Continuity

Ex.869Continuous extension

#continuous extension #removable singularity

Continuity

High school

Ex.1071Boundary effects

#boundary #integration bounds #evaluation

Multiple integrals

Ex.1072Cartesian vs polar computation

#polar vs cartesian #comparison #disk

Multiple integrals

Ex.1073Double integral over an annulus

#polar coordinates #annulus #angular integral

Multiple integrals

Ex.1074Double integral over a disk tangent at origin

#polar coordinates #circle offset #integral

Multiple integrals

Ex.1075Double integral over a disk in polar coordinates

#polar coordinates #disk #Jacobian

Multiple integrals

Ex.1076Polar integral over a half-disk

#polar #half-disk #integral

Multiple integrals

Ex.1077Mass of a cube with variable density

#triple integral #density #cube

Multiple integrals

Ex.1078Triple integral over a simplex

#triple integral #simplex #iterated

Multiple integrals

Ex.1079General chain rule

#chain rule #composition #multivariable

Partial derivatives

Ex.1080Clairaut's theorem

#Clairaut #Schwarz #mixed partials

Partial derivatives

Ex.1081Saddle point

#saddle point #Hessian #classification

Optimisation in several variables

Ex.1082The differential

#differential #linearization #approximation

Partial derivatives

Ex.1083Second-order partial derivatives

#higher order #Schwarz #second derivatives

Partial derivatives

Ex.1084The Laplacian

#Laplacian #harmonic functions #second derivatives

Partial derivatives

Ex.1085Finding critical points

#critical points #stationary #gradient

Partial derivatives

Ex.1086Schwarz's theorem and a counterexample

#Schwarz theorem #mixed partials #symmetry

Partial derivatives

Ex.1087Tangent plane to a graph

#tangent plane #differentiability #graph

Partial derivatives

Ex.1198The curl

#curl #vector field #rotation

Vector calculus

Ex.1201Flux through a flat surface

#flux #plane surface #dot product

Vector calculus

Ex.1202An application of Green's theorem

#Green #application #area

Vector calculus

Ex.1203Area using Green's theorem

#area #Green #closed curve

Vector calculus

Ex.1204Green with symmetry

#Green #symmetry #shortcut

Vector calculus

Ex.1210Conservative vector field

#conservative #gradient #path independence

Vector calculus

Ex.1211Fundamental theorem for line integrals

#fundamental theorem #line integral #potential

Vector calculus

Ex.1212Line integral along a circle

#line integral #circle #parametrization

Vector calculus

Ex.1213Line integral on a piecewise path

#piecewise path #line integral #work

Vector calculus

Ex.1214Work done by a force field

#work #line integral #vector field

Vector calculus

Ex.1215Testing whether a field is conservative

#conservative #curl #test

Vector calculus

Ex.1475Cauchy-Schwarz inequality via optimisation

#Cauchy-Schwarz #optimisation #proof

Optimisation in several variables

Olympiad

Ex.1476Constrained extrema and a compactness argument

#Lagrange #compactness #global maximum

Optimisation in several variables

Ex.1477Maximizing a product under a linear constraint

#Lagrange #product #linear constraint

Optimisation in several variables

Ex.1478Least squares as an optimisation problem

#least squares #optimisation #regression

Optimisation in several variables

Ex.1497Affine change of variables

#change of variables #Jacobian #parallelogram

Multiple integrals

Ex.1498Is a function $C^1$ on $\mathbb{R}^2$?

#continuity #partial derivatives #class C1

Partial derivatives

Ex.1518Divergence theorem (Ostrogradsky)

#divergence theorem #flux #surface

Vector calculus

Ex.1519Green-Riemann formula

#Green-Riemann #line integral #double integral

Vector calculus

Ex.1520Stokes' theorem

#Stokes #curl #surface integral

Vector calculus

Ex.1591Cauchy–Schwarz via optimization

#Cauchy-Schwarz #optimization #inequality

Optimisation in several variables

Ex.1592Optimization under inequality constraints

#inequality constraint #KKT #boundary

Optimisation in several variables

Ex.1593Classifying a 3D critical point

#Hessian #3D #saddle

Optimisation in several variables

Ex.1594Three-variable saddle point

#three variables #saddle #Hessian

Optimisation in several variables

Ex.1595Shortest distance from a point to a line

#Lagrange #distance #line

Optimisation in several variables

Ex.1596Distance from origin to an ellipsoid

#Lagrange #ellipsoid #distance

Optimisation in several variables

Ex.1597Lagrange with exponential objective

#Lagrange #exponential #constrained maximum

Optimisation in several variables

Ex.1598Global extrema on a compact set

#compactness #global extrema #boundary

Optimisation in several variables

Ex.1599Maximizing a volume with fixed perimeter

#Lagrange #geometric optimization #volume

Optimisation in several variables

Ex.1600Constrained quadratic form

#quadratic form #Rayleigh quotient #eigenvalues

Optimisation in several variables

Ex.1601Image of a compact set under a continuous function

#compactness #continuous image #extrema

Continuity

Ex.1602A function that is not continuous despite radial limits

#continuity #radial limit #counterexample

Continuity

Ex.1783Centre of gravity of a planar region

#centroid #parabola #uniform density

Multiple integrals

Ex.1784Linear change of variables

#change of variables #Jacobian #triangle

Multiple integrals

Ex.1785Change of variables to simplify the integrand

#change of variables #Jacobian #triangular

Multiple integrals

Ex.17863D change of variables

#change of variables #3D #Jacobian

Multiple integrals

Ex.1787Volume using cylindrical coordinates

#cylindrical coordinates #volume #cone

Multiple integrals

Ex.1788The Fubini trick

#Fubini #double integral #iterated

Multiple integrals

Ex.1789Polar region between two circles

#polar #circle #region between circles

Multiple integrals

Ex.1790Spherical coordinates with a radial integrand

#spherical coordinates #radial #integration

Multiple integrals

Ex.1791Volume of a sphere by triple integral

#spherical coordinates #volume #sphere

Multiple integrals

Ex.1792Triple integral over a cube

#cube #triple integral #iterated

Multiple integrals

Ex.1793Triple integral over a simplex with cube

#simplex #triple integral #iterated

Multiple integrals

Ex.1794Triple integral with sine

#triple integral #sine #simplex

Multiple integrals

Ex.1795Differentiability of a piecewise function

#differentiability #continuity #piecewise

Partial derivatives

Ex.1796The Hessian matrix

#Hessian #second derivatives #quadratic form

Optimisation in several variables

Ex.1797Implicit differentiation

#implicit differentiation #Implicit Function Theorem

Partial derivatives

Ex.1798Inverse function theorem (illustration)

#inverse function theorem #Jacobian #local invertibility

Partial derivatives

Ex.1799Jacobian of a composition

#Jacobian #composition #chain rule

Partial derivatives

Ex.1800Level curves and gradients

#level curves #gradient #orthogonality

Partial derivatives

Ex.1801Mean value inequality

#mean value #inequality #gradient

Partial derivatives

Ex.1802Taylor expansion of order 2

#Taylor expansion #multivariable #approximation

Partial derivatives

Ex.1903Gauss divergence theorem for a cylinder

#divergence #cylinder #flux

Vector calculus

Ex.1904The divergence theorem

#divergence theorem #flux #surface

Vector calculus

Ex.1905Flux through a cube

#divergence #cube #flux

Vector calculus

Ex.1908Divergence theorem with variable divergence

#divergence #sphere #variable density

Vector calculus

Ex.1910Recovering a field from its curl (2D)

#Green #curl #reconstruction

Vector calculus

Ex.1911Green's reciprocity

#Green #reciprocity #symmetry

Vector calculus

Ex.1912Green's theorem on an annulus

#Green #annulus #inner boundary

Vector calculus

Ex.1913Green's theorem

#Green theorem #curl #region

Vector calculus

Ex.1914Green's theorem with a general field

#Green #general field #region

Vector calculus

Ex.1917Area of a region using line integral

#line integral #area #polar

Vector calculus

Ex.1918Conservative fields in 3D

#conservative #3D #potential

Vector calculus

Ex.1919Finding a potential function

#potential #conservative #integration

Vector calculus

Ex.1925Stokes with a hemisphere

#Stokes #hemisphere #closed curve

Vector calculus

Ex.1926Stokes' theorem

#Stokes theorem #curl #surface

Vector calculus

Ex.2070Rayleigh quotient and eigenvalues

#Rayleigh quotient #eigenvalues #optimisation on the sphere

Optimisation in several variables

Olympiad

Ex.2071The AM-GM inequality via optimisation

#inequality #AM-GM #optimisation #proof

Optimisation in several variables

Olympiad

Ex.2115Two equality constraints

#Lagrange #multiple constraints #KKT

Optimisation in several variables

Ex.2183Gaussian integral over $\mathbb{R}^2$

#Gaussian #polar #improper integral

Multiple integrals

Ex.2184Jacobian of a nonlinear transformation

#Jacobian #nonlinear #change of variables

Multiple integrals

Ex.2185Harmonic conjugate

#harmonic conjugate #Cauchy-Riemann #complex

Partial derivatives

Ex.2186Partial derivatives exist but function is not differentiable

#differentiability #counterexample #partial derivatives

Partial derivatives

Ex.2223Stokes on a cone

#Stokes #cone #surface

Vector calculus

Ex.2224Stokes on a torus (statement)

#Stokes #torus #closed surface

Vector calculus