Proof of the volume of a ball in R^3 from the n-dimensional formula

Builds on Volume of a ball in R^n Gamma function The gamma function extends the factorial Gaussian integral Integration by substitution

Proof

We take the case $n = 3$ of the general formula.

Let $R > 0$ and $B = \set{(x, y, z) \in \R^3 : x^2 + y^2 + z^2 \le R^2}$ . This is the closed ball $B_3(R)$ of radius $R$ about $0$ .
By the volume of a ball in R^n with $n = 3$ , $\operatorname{vol}(B) = \pi^{3/2} R^3 / \Gamma(5/2)$ , where $\Gamma$ is the gamma function.
We compute $\Gamma(1/2)$ . For $0 < \varepsilon < M$ , the substitution $t = u^2$ gives $\int_\varepsilon^M t^{-1/2} e^{-t} \, dt = 2 \int_{\sqrt{\varepsilon}}^{\sqrt{M}} e^{-u^2} \, du$ .
Let $\varepsilon \to 0$ and $M \to \infty$ . The integrand $e^{-u^2}$ is even, so the Gaussian integral gives $\Gamma(1/2) = \int_{-\infty}^{\infty} e^{-u^2} \, du = \sqrt{\pi}$ .
By the functional equation $\Gamma(s + 1) = s \, \Gamma(s)$ , applied twice, $\Gamma(5/2) = \tfrac{3}{2} \cdot \tfrac{1}{2} \cdot \Gamma(1/2) = \tfrac{3}{4} \sqrt{\pi}$ .
Hence $\operatorname{vol}(B) = \pi^{3/2} R^3 \cdot \frac{4}{3 \sqrt{\pi}} = \frac{4}{3} \pi R^3$ .

Idea

The volume of a ball in every dimension is given by one formula with the gamma function. In dimension $3$ the gamma value $\Gamma(5/2)$ reduces to $\Gamma(1/2) = \sqrt{\pi}$ , and the powers of $\pi$ combine to give $\tfrac{4}{3} \pi R^3$ .

Reading path

To understand this page from the ground up, read these 646 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of integration by parts, Proof of the Cauchy–Hadamard theorem, Proof that a function with nonnegative second derivative is convex, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the decomposition of a permutation into transpositions, Proof that the symmetric group is generated by transpositions, Proof of the formula for the derivative of a polynomial, Proof of the Laplace expansion along a column by transposition, Proof that open sets are the sets that are neighbourhoods of all their points, Proof that closed balls are closed, Proof of the diameter characterisation of Cauchy sequences, Proof that exponential growth beats polynomial growth, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of term-by-term differentiation, Proof of the epsilon-delta form of continuity, Proof of Fubini's theorem for Riemann integrals on a rectangle, Proof that equivalence classes form a partition, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule in several variables, Proof of the chain rule with an auxiliary difference quotient, Proof of the integral of the exponential function, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof of the reverse triangle inequality, Proof that the determinant is multiplicative by expanding the rows, Proof that permuting rows multiplies the determinant by the sign, Proof that the exponential series has infinite radius of convergence, Proof that closed subsets of complete spaces are complete, Proof of the derivative as a little-o approximation, Proof that disjoint cycles commute, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a matrix is invertible iff its determinant is a unit, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof that a function with nonnegative derivative is increasing, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of continuity via closed sets, Proof that a matrix with two equal rows has determinant zero, Proof of the double negation law by a truth table, Proof that the logarithm is the inverse of the exponential, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof of the derivative of the logarithm by the inverse function rule, Proof of the Laplace expansion along a row by the Leibniz formula, Proof that compactness and sequential compactness agree in metric spaces, Proof that the exponential function is positive, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that Lipschitz functions are uniformly continuous, Proof that the integral of a continuous function is an antiderivative, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that linear maps on finite-dimensional normed spaces are continuous, Proof that power series are differentiable inside the radius of convergence, Proof that the determinant of a transpose equals the determinant, Proof of the laws of exponents in a group, Proof of the limit of a product of sequences, Proof of the inverse in terms of the adjugate, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, Proof of the chain rule for a composition with a curve, Proof that a differentiable function is convex iff its derivative is increasing, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof that a continuous image of a compact set is compact, Proof of the Heine–Cantor theorem by a finite cover of balls, Proof that compact subsets of a Hausdorff space are closed, Proof that the composition of continuous maps is continuous, Proof of epsilon-delta continuity at a point in metric spaces, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof that the exponential function is strictly increasing, Proof that Cauchy sequences are bounded, Proof of the description of the open sets of a metric space, Proof of the Banach fixed point theorem by iteration, Proof of the triangle inequality for real numbers, Proof that an intersection of subgroups is a subgroup, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof that the sign is a homomorphism, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the volume of a ball in R^n by slicing, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof of the determinant of the inverse, Proof that every spanning set contains a basis, by Zorn's lemma, Proof that the determinant of the identity matrix is one, Proof of the continuity of the inverse of a strictly monotone continuous function, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the inverse function theorem by the contraction mapping principle, Proof that closed sets contain their limits, Proof that 1 + x is at most exp x, Proof of the linearity of the integral, Proof of the factorial formula for binomial coefficients by induction, Proof that a function with positive derivative is strictly increasing, Proof of integration by substitution, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof that the derivative gives the directional derivatives, Proof of the second fundamental theorem of calculus, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof of the decomposition of a permutation into disjoint cycles, Proof that one-sided limits of a monotone function exist, Proof of the product of a matrix and its adjugate by Laplace expansion, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the derivative of x^a by the chain rule, Proof of the division algorithm, Proof of the Archimedean property in an Archimedean ordered field, Proof of the sign of a cycle, Proof that finite-dimensional normed spaces are complete, Proof that a continuous real function on a compact set attains its bounds, Proof of the Leibniz formula from the defining properties of the determinant, Proof of the uniqueness of inverses, Proof of the derivative of a map into a product, Proof that a composition of continuous functions is continuous, Proof of the exponential of a sum by the Cauchy product, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the exponential function is its own derivative, by term-by-term differentiation, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that the determinant is multilinear in the rows, Proof that open balls are open, Proof that the gamma function extends the factorial, Proof of the monotonicity of the integral, Proof of the comparison test for improper integrals, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof of the ratio formula for the radius of convergence, Proof that C^k functions are characterised by continuous derivatives, Proof that continuous partial derivatives imply differentiability, Proof that a set is open if and only if its complement is closed, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the subgroup criterion, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof that slopes of a convex function are increasing, Proof that sums and products of continuous functions are continuous, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the mean value inequality, Proof that the exponential function is strictly convex, Proof of the constant multiple rule.

Affine combination concept
Basis concept
Binary operation definition
Binomial coefficient definition
Borel sigma-algebra concept
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Convex function definition
Coordinate space object
Coordinate vector concept
Cycle notation concept
Definition by recursion concept
Divisibility definition
Element definition
Euclidean metric example
Euclidean space object
Extended real number line object
Factorial definition
Function concept
Function composition concept
Group concept
Identity matrix object
Imaginary unit object
Inner product concept
Inner product space concept
Integers object
Integrable function concept
Integral of a non-negative function concept
Limit point of a set definition
Linear isomorphism concept
Linear map concept
Lipschitz continuity definition
Local maximum definition
Mathematical induction concept
Matrix definition
Multilinear map concept
Natural numbers object
Norm concept
Norm induced by an inner product concept
Ordered field definition
Outer measure concept
Partition of an interval definition
Peano axioms concept
Permutation concept
Permutation definition
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Strictly monotone function definition
Transposition concept
Vector space definition
p-metric on R^n example
Identity element definition
Proof of Pascal's rule from the recursive definition proof
Real and imaginary parts definition
Convex set definition
Proof that slopes of a convex function are increasing proof
Strictly convex function definition
Set definition
Fréchet derivative definition
Having a directional derivative definition
Total derivative concept
Function of several variables concept
Vector-valued function concept
Lebesgue measure object
Group homomorphism definition
Ring definition
Subgroup definition
Symmetric group definition
Ring of integers statement
Lebesgue integral definition
Limit of a function definition
Local minimum definition
Matrix multiplication definition
Square matrix definition
Transpose definition
Archimedean ordered field definition
Successor function definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Inner product definition
Linear combination definition
Linear map definition
Linear subspace definition
Norm definition
p-norm on R^n example
Inverse element definition
Pascal's rule statement
Complex conjugate definition
Empty set object
Family of sets definition
Ordered pair definition
Slopes of a convex function are increasing statement
Differentiable function of several variables concept
Iterated integral definition
Partial derivative concept
Limit of a function of several variables concept
Parametrised curve concept
Transposition definition
Commutative ring definition
Unit of a ring definition
Intersection definition
Union definition
Proof that an intersection of subgroups is a subgroup proof
Continuity at a point definition
Derivative definition
Limit at infinity definition
One-sided limit definition
Set-builder notation concept
Invertible matrix definition
Addition of natural numbers definition
Riemann integrable function definition
Biconditional definition
Truth table concept
Modus ponens concept
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
Existential quantifier definition
Universal quantifier definition
The successor function is injective statement
Zero is not a successor statement
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Sequence tending to infinity definition
Series definition
Inner product space definition
Linear independence definition
Composition of linear maps definition
Linear isomorphism definition
Linear span definition
Multilinear map definition
Normed vector space definition
Completeness axiom statement
Cartesian product definition
Proof of the factorial formula for binomial coefficients by induction proof
Gradient concept
Proof of the equality of complex numbers proof
Disjoint sets definition
Jacobian matrix concept
Proof of the uniqueness of inverses proof
Partial derivative definition
Continuity in R^n definition
Differentiable function of several variables definition
Union of a family of sets definition
Continuous function definition
Antiderivative definition
Differentiable function definition
Axiom of extensionality statement
One-sided derivative definition
Subset definition
Riemann integral definition
Archimedean property statement
Proof of the derivative as a little-o approximation proof
Proof of the commutativity of addition by induction proof
Mathematical proof concept
Order on the natural numbers definition
Uniqueness quantifier definition
Propositional logic concept
Limit superior definition
The rationals form an ordered field statement
Proof of the sum of a finite geometric series proof
Algebra of limits concept
Convergent sequence definition
Pointwise convergence definition
Sequential continuity definition
The sequence r^n tends to infinity for r > 1 example
Upper and lower Darboux sums definition
Cauchy product definition
Partial sum definition
Power series definition
Norm induced by an inner product definition
Spanning set definition
Alternating multilinear map definition
Equality of complex numbers statement
Higher partial derivatives definition
Factorial formula for binomial coefficients statement
Binary relation definition
Differentiable on an open set definition
Double integral definition
Jacobian matrix definition
Uniqueness of inverses statement
Partition of a set definition
Topological space definition
Universal set definition
Proof of the derivative of a map into R^n by components proof
Derivative as a little-o approximation statement
Proof that a composition of continuous functions is continuous proof
Derivative function definition
Order on the reals definition
Definite integral definition
Improper integral definition
Tautology definition
Commutativity of addition statement
Determinant definition
Proof of strong induction from ordinary induction proof
Uniform convergence definition
Sum of a finite geometric series statement
Upper and lower Darboux integrals definition
Convergent series definition
Radius of convergence definition
Finite-dimensional vector space definition
Basis definition
Antisymmetric relation definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Iterated derivative definition
Proof of the binomial theorem by induction proof
Jacobian determinant definition
Sequential compactness definition
Function definition
Complement definition
Subspace topology definition
Higher derivative definition
Proof of the laws of exponents in a group proof
Proof of the subgroup criterion proof
Interval definition
Proof that a matrix with two equal rows has determinant zero proof
Proof that the determinant of the identity matrix is one proof
Uniform convergence on a set definition
Logical equivalence definition
Strong induction statement
Minor definition
Coordinate vector definition
Proof that the determinant is multilinear in the rows proof
Sum of a series definition
Series of functions definition
Absolute convergence definition
Equivalence relation definition
Preorder definition
Binomial theorem statement
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Choice function definition
Second derivative definition
Continuously differentiable function definition
Image of a set definition
Preimage of a set definition
Restriction of a function definition
Subgroup criterion statement
Proof of the epsilon-delta form of continuity proof
Laws of exponents in a group statement
Proof of the epsilon-delta form of the limit of a function proof
Determinant of a matrix with two equal rows statement
Metric space definition
Determinant of the identity matrix statement
Cofactor definition
Determinant is multilinear in the rows statement
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Left inverse definition
Monotone function definition
Right inverse definition
Surjective function definition
Axiom of choice statement
Proof that disjoint cycles commute proof
Equivalence class definition
Intersection of subgroups is a subgroup statement
Subspace metric definition
Epsilon-delta definition of continuity statement
Epsilon-delta definition of a limit statement
Closed ball definition
Convergence of a sequence in a metric space definition
Lipschitz map definition
Open ball definition
Triangle inequality for a metric statement
Adjugate matrix definition
Diameter of a set definition
Double negation law statement
Well-ordering principle statement
Maximal element definition
Total order definition
Upper bound definition
Bijective function definition
Proof that equivalence classes form a partition proof
Uniform continuity definition
Disjoint cycles commute statement
Composition of continuous functions is continuous statement
Subgroup generated by a subset definition
Metric topology definition
Bounded set definition
Contraction mapping definition
Proof of the division algorithm proof
Proof by contradiction concept
Proof of the diameter characterisation of Cauchy sequences proof
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Proof that a function is bijective iff it has a two-sided inverse proof
Uniform continuity on a set definition
Equivalence classes form a partition statement
Open set definition
Division algorithm statement
Epsilon characterisation of Cauchy sequences statement
Proof that Lipschitz functions are uniformly continuous proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Bijective iff it has a two-sided inverse statement
Hausdorff space definition
Open cover definition
Closed set definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Continuous map of topological spaces definition
Proof that the derivative gives the directional derivatives proof
Proof of the decomposition of a permutation into disjoint cycles proof
Equivalent metrics definition
Zorn's lemma statement
Proof of the description of the open sets of a metric space proof
Complete metric space definition
Lipschitz functions are uniformly continuous statement
The reals form an ordered field statement
Compact space definition
Proof that a set is open if and only if its complement is closed proof
Proof that the composition of continuous maps is continuous proof
Open sets characterised by neighbourhoods statement
Derivative gives the directional derivatives statement
Equivalent norms definition
Decomposition of a permutation into disjoint cycles statement
Open sets of a metric space statement
Absolute value definition
Proof that the coordinate map is a linear isomorphism proof
Proof that every spanning set contains a basis, by Zorn's lemma proof
Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Proof of the existence of square roots of nonnegative reals proof
Complement of an open set is closed statement
Composition of continuous maps is continuous statement
Compact set definition
Continuous linear map definition
Continuity at a point of a topological space definition
Banach space definition
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that closed sets contain their limits proof
Proof of the decomposition of a permutation into transpositions proof
Proof that closed balls are closed proof
Proof that open balls are open proof
Proof of the epsilon-N definition of the limit of a sequence proof
Isomorphism with the coordinate space statement
Every spanning set contains a basis statement
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Existence of the infimum statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Proof of continuity via closed sets proof
Triangle inequality statement
Absolute value is multiplicative statement
Closed sets contain their limits statement
Decomposition of a permutation into transpositions statement
Closed balls are closed statement
Open balls are open statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof that one-sided limits of a monotone function exist proof
Proof of the intermediate value theorem by the supremum proof
Proof that the reals are Archimedean from the completeness axiom proof
Compactness via open covers statement
Continuity via closed sets statement
Modulus of a complex number definition
Proof of the reverse triangle inequality proof
Sign of a permutation definition
Proof of the limit of a sum of sequences proof
Proof that closed subsets of complete spaces are complete proof
Proof of the uniqueness of limits proof
Proof of the squeeze theorem for sequences proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof that limits preserve non-strict inequalities proof
Proof of continuity via open sets in metric spaces proof
Intermediate value theorem statement
Proof that the symmetric group is generated by transpositions proof
Proof that metric spaces are Hausdorff proof
Proof that differentiable functions are continuous proof
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
The sequence 1/n tends to zero example
Proof that Cauchy sequences are bounded proof
The reals are Archimedean statement
Proof of the term test proof
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
Reverse triangle inequality statement
Proof that compact subsets of a Hausdorff space are closed proof
Proof that a complex number times its conjugate is the square of its modulus proof
Proof that a continuous image of a compact set is compact proof
Closed subsets of complete spaces are complete statement
Lebesgue number lemma statement
Epsilon-delta continuity in metric spaces statement
The symmetric group is generated by transpositions statement
Proof that compact subsets of a metric space are bounded proof
Proof that the sign is a homomorphism proof
Proof of the sign of a cycle proof
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
Differentiable functions are continuous statement
Limit of a sum of sequences statement
Proof of the sequence characterisation of uniform continuity proof
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Proof of the continuity of the inverse of a strictly monotone continuous function proof
Proof of the triangle inequality for norms proof
Linearity of the integral statement
Proof of the ratio formula for the radius of convergence proof
Convergent sequences are bounded statement
Squeeze theorem statement
Proof of the monotonicity of the integral proof
Proof that there is a natural number n with 1/n below any positive epsilon proof
Subsequences of a convergent sequence converge to the same limit statement
Convergent sequences are Cauchy statement
Proof that Darboux and Riemann integrals agree proof
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Compact subsets of a Hausdorff space are closed statement
A complex number times its conjugate statement
Continuous image of a compact set is compact statement
Sign of a cycle statement
Compactness and sequential compactness agree in metric spaces statement
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
Sign is a homomorphism statement
Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
One-sided limits of a monotone function exist statement
Proof that permuting rows multiplies the determinant by the sign proof
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
There is a natural number n with 1/n below any positive epsilon statement
Term test statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Triangle inequality for norms statement
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof of the Heine–Cantor theorem by a finite cover of balls proof
Proof that Lipschitz maps are uniformly continuous proof
Proof of the Banach fixed point theorem by iteration proof
Permuting rows multiplies the determinant by the sign statement
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Proof of the Cauchy–Schwarz inequality proof
Limit of a product of sequences statement
Continuity of the inverse of a strictly monotone continuous function statement
Decreasing sequence bounded below converges statement
Geometric series example
A series of nonnegative terms converges iff its partial sums are bounded statement
Proof of Riemann's criterion for integrability proof
Heine–Cantor theorem statement
Banach fixed point theorem statement
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Riemann's criterion for integrability statement
Sequential characterization of limits statement
Proof of the Leibniz formula from the defining properties of the determinant proof
Proof of the root test proof
Proof of the Cauchy–Hadamard theorem proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the ratio test proof
Proof of the comparison test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Cauchy–Schwarz inequality statement
Bounded linear maps are continuous statement
Comparison test statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof that sums and products of continuous functions are continuous proof
Proof of the product rule proof
Leibniz formula for determinants statement
Proof of the linearity of differentiation proof
Cauchy product of absolutely convergent series statement
Bolzano–Weierstrass theorem statement
Exponentials dominate powers in sequences statement
Proof of the additivity of the integral over intervals proof
Product of continuous functions is continuous statement
Proof of the derivative of a map into a product proof
Chain rule statement
Product rule statement
Proof of the Heine–Borel theorem by sequential compactness proof
Ratio test statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Additivity of the integral over intervals statement
Linearity of differentiation statement
Operator norm definition
Root test statement
Proof of the Laplace expansion along a row by the Leibniz formula proof
Proof that the determinant is multiplicative by expanding the rows proof
Proof that the determinant of a transpose equals the determinant proof
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Completeness of the real numbers statement
Derivative of a map into a product statement
Continuous functions on a closed bounded interval are uniformly continuous statement
Heine–Borel theorem statement
Proof of the constant multiple rule proof
Derivative of an inverse function statement
Boundedness theorem statement
Proof of the power rule by induction proof
Laplace expansion along a row statement
Proof of the first fundamental theorem of calculus proof
Ratio formula for the radius of convergence statement
Cauchy–Hadamard theorem statement
Determinant is multiplicative statement
Determinant of a transpose statement
Proof of the comparison test for improper integrals proof
Cauchy criterion for sequences statement
Constant multiple rule statement
Derivative of a map into R^n by components statement
Power rule statement
Comparison test for improper integrals statement
Proof of the Laplace expansion along a column by transposition proof
Proof of the determinant of the inverse proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Proof that a continuous real function on a compact set attains its bounds proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Extreme value theorem statement
Gamma function definition
Proof of the formula for the derivative of a polynomial proof
Continuous real function on a compact set attains its bounds statement
Laplace expansion along a column statement
Determinant of the inverse statement
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof that continuous functions are Riemann integrable proof
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Proof that absolutely convergent series converge proof
Proof of the product of a matrix and its adjugate by Laplace expansion proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Equivalence of norms in finite dimension statement
Absolutely convergent series converge statement
Product of a matrix and its adjugate statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of Fubini's theorem for Riemann integrals on a rectangle proof
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Proof that finite-dimensional normed spaces are complete proof
Proof of the mean value theorem for integrals proof
Proof of the inverse in terms of the adjugate proof
Proof of the Weierstrass M-test proof
Linear maps on finite-dimensional spaces are continuous statement
Finite-dimensional normed spaces are complete statement
Fubini's theorem statement
Mean value theorem for integrals statement
Proof of the mean value theorem from Rolle's theorem proof
Weierstrass M-test statement
Inverse in terms of the adjugate statement
Mean value theorem statement
Tonelli's theorem statement
First fundamental theorem of calculus statement
Proof of the chain rule in several variables proof
Proof that a matrix is invertible iff its determinant is a unit proof
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Proof that a function with nonnegative derivative is increasing proof
Proof that continuous partial derivatives imply differentiability proof
Proof that the integral of a continuous function is an antiderivative proof
Chain rule in several variables statement
Proof that a differentiable function is convex iff its derivative is increasing proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
A matrix is invertible iff its determinant is a unit statement
Power series converge uniformly on smaller discs inside the radius of convergence statement
Proof that a function with positive derivative is strictly increasing proof
Proof of the second fundamental theorem of calculus proof
A function with nonnegative derivative is increasing statement
The integral of a continuous function is an antiderivative statement
A differentiable function is convex iff its derivative is increasing statement
Derivative of the inverse function in several variables statement
Continuous partial derivatives imply differentiability statement
Proof of the chain rule for a composition with a curve proof
Uniform convergence of derivatives allows term-by-term differentiation statement
Second fundamental theorem of calculus statement
Proof that the exponential series has infinite radius of convergence proof
A function with positive derivative is strictly increasing statement
Chain rule for a composition with a curve statement
Proof that a function with nonnegative second derivative is convex proof
Proof of integration by parts proof
Proof that C^k functions are characterised by continuous derivatives proof
Proof of integration by substitution proof
The exponential series has infinite radius of convergence statement
Proof of term-by-term differentiation proof
C^k function characterised by continuous derivatives statement
A function with nonnegative second derivative is convex statement
Integration by substitution statement
Integration by parts statement
Proof of the mean value inequality proof
Term-by-term differentiation statement
Exponential function definition
Proof that exponential growth beats polynomial growth proof
Gaussian integral example
Proof of the exponential of a sum by the Cauchy product proof
Mean value inequality statement
Proof that power series are differentiable inside the radius of convergence proof
Exponential of a sum statement
Proof of the inverse function theorem by the contraction mapping principle proof
Exponential growth beats polynomial growth statement
Power series are differentiable inside the radius of convergence statement
Proof that the exponential function is positive proof
Inverse function theorem statement
Proof that the exponential function is its own derivative, by term-by-term differentiation proof
The exponential function is positive statement
Diffeomorphism definition
The exponential function is its own derivative statement
Change of variables in multiple integrals statement
Proof that the exponential function is strictly increasing proof
Proof of the integral of the exponential function proof
The exponential function is strictly increasing statement
Integral of the exponential function statement
Proof that the exponential function is strictly convex proof
Proof that 1 + x is at most exp x proof
Natural logarithm definition
Integral of exp(-x) over the half line example
The exponential function is convex statement
Real exponentiation definition
1 + x is at most exp x statement
Proof that the logarithm is the inverse of the exponential proof
Logarithm is the inverse of the exponential statement
Proof of the derivative of the logarithm by the inverse function rule proof
Derivative of the logarithm statement
Proof of the derivative of x^a by the chain rule proof
Derivative of x^a statement
Proof that the gamma function extends the factorial proof
The gamma function extends the factorial statement
Proof of the volume of a ball in R^n by slicing proof
Volume of a ball in R^n statement
Proof of the volume of a ball in R^3 from the n-dimensional formula proof

Where this leads

1 page builds on this one.

Volume of a ball in R^3

Tag	`Q89SKMVRMB`
URI	`m2://Q89SKMVRMB`
Kind	proof
Writing	W0 stub
MSC	26B15
Proves	Volume of a ball in R^3
Uses	Volume of a ball in R^n, Gamma function, The gamma function extends the factorial, Gaussian integral, Integration by substitution
Changed	4 November 2026

Proof of the volume of a ball in R^3 from the n-dimensional formula

Proof

Idea

Reading path

Where this leads

Further reading

What this page needs for W1

What links here