It has been suggested that this article be merged into Multivariable calculus. (Discuss) Proposed since May 2022.

In mathematics, calculus on Euclidean space is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space as well as a finite-dimensional real vector space. This calculus is also known as advanced calculus, especially in the United States. It is similar to multivariable calculus but is somehow more sophisticated in that it uses linear algebra (or some functional analysis) more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of differential forms. This extensive use of linear algebra also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces.

Calculus on Euclidean space is also a local model of calculus on manifolds, a theory of functions on manifolds.

Basic notions

See also: Function of a real variable and Multivariable calculus

Functions in one real variable

This section is a brief review of function theory in one-variable calculus.

A real-valued function is continuous at if it is approximately constant near ; i.e.,

In contrast, the function is differentiable at if it is approximately linear near ; i.e., there is some real number such that

[1]

(For simplicity, suppose . Then the above means that where goes to 0 faster than h going to 0 and, in that sense, behaves like .)

The number depends on and thus is denoted as . If is differentiable on an open interval and if is a continuous function on , then is called a C1 function. More generally, is called a Ck function if its derivative is Ck-1 function. Taylor's theorem states that a Ck function is precisely a function that can be approximated by a polynomial of degree k.

If is a C1 function and for some , then either or ; i.e., either is strictly increasing or strictly decreasing in some open interval containing a. In particular, is bijective for some open interval containing . The inverse function theorem then says that the inverse function is differentiable on U with the derivatives: for

Derivative of a map and chain rule

For functions defined in the plane or more generally on an Euclidean space , it is necessary to consider functions that are vector-valued or matrix-valued. It is also conceptually helpful to do this in an invariant manner (i.e., a coordinate-free way). Derivatives of such maps at a point are then vectors or linear maps, not real numbers.

Let be a map from an open subset of to an open subset of . Then the map is said to be differentiable at a point in if there exists a (necessarily unique) linear transformation , called the derivative of at , such that

where is the application of the linear transformation to .[2] If is differentiable at , then it is continuous at since

as .

As in the one-variable case, there is

Chain rule — [3] Let be as above and a map for some open subset of . If is differentiable at and differentiable at , then the composition is differentiable at with the derivative

This is proved exactly as for functions in one variable. Indeed, with the notation , we have:

Here, since is differentiable at , the second term on the right goes to zero as . As for the first term, it can be written as:

Now, by the argument showing the continuity of at , we see is bounded. Also, as since is continuous at . Hence, the first term also goes to zero as by the differentiability of at .

The map as above is called continuously differentiable or if it is differentiable on the domain and also the derivatives vary continuously; i.e., is continuous.

Corollary — If are continuously differentiable, then is continuously differentiable.

As a linear transformation, is represented by an -matrix, called the Jacobian matrix of at and we write it as:

Taking to be , a real number and the j-th standard basis element, we see that the differentiability of at implies:

where denotes the i-th component of . That is, each component of is differentiable at in each variable with the derivative . In terms of Jacobian matrices, the chain rule says ; i.e., as ,

which is the form of the chain rule that is often stated.

A partial converse to the above holds. Namely, if the partial derivatives are all defined and continuous, then is continuously differentiable.[4] This is a consequence of the mean value inequality:

Mean value inequality — [5] Given the map as above and points in such that the line segment between lies in , if is continuous on and is differentiable on the interior, then, for any vector ,

where

(This version of mean value inequality follows from mean value inequality in mean value theorem#Mean value theorem for vector-valued functions applied to the function , where the proof on mean value inequality is given.)

Indeed, let . We note that, if , then

For simplicity, assume (the argument for the general case is similar). Then, by mean value inequality, with the operator norm ,

which implies as required.

Example: Let be the set of all invertible real square matrices of size n. Note can be identified as an open subset of with coordinates . Consider the function = the inverse matrix of defined on . To guess its derivatives, assume is differentiable and consider the curve where means the matrix exponential of . By the chain rule applied to , we have:

.

Taking , we get:

.

Now, we then have:[6]

Since the operator norm is equivalent to the Euclidean norm on (any norms are equivalent to each other), this implies is differentiable. Finally, from the formula for , we see the partial derivatives of are smooth (infinitely differentiable); whence, is smooth too.

Higher derivatives and Taylor formula

If is differentiable where is an open subset, then the derivatives determine the map , where stands for homomorphisms between vector spaces; i.e., linear maps. If is differentiable, then . Here, the codomain of can be identified with the space of bilinear maps by:

where and is bijective with the inverse given by .[a] In general, is a map from to the space of -multilinear maps .

Just as is represented by a matrix (Jacobian matrix), when (a bilinear map is a bilinear form), the bilinear form is represented by a matrix called the Hessian matrix of at ; namely, the square matrix of size such that , where the paring refers to an inner product of , and is none other than the Jacobian matrix of . The -th entry of is thus given explicitly as .

Moreover, if exists and is continuous, then the matrix is symmetric, the fact known as the symmetry of second derivatives.[7] This is seen using the mean value inequality. For vectors in , using mean value inequality twice, we have:

which says

Since the right-hand side is symmetric in , so is the left-hand side: . By induction, if is , then the k-multilinear map is symmetric; i.e., the order of taking partial derivatives does not matter.[7]

As in the case of one variable, the Taylor series expansion can then be proved by integration by parts:

Taylor's formula has an effect of dividing a function by variables, which can be illustrated by the next typical theoretical use of the formula.

Example:[8] Let be a linear map between the vector space of smooth functions on with rapidly decreasing derivatives; i.e., for any multi-index . (The space is called a Schwartz space.) For each in , Taylor's formula implies we can write:

with , where is a smooth function with compact support and . Now, assume commutes with coordinates; i.e., . Then

.

Evaluating the above at , we get In other words, is a multiplication by some function ; i.e., . Now, assume further that commutes with partial differentiations. We then easily see that is a constant; is a multiplication by a constant.

(Aside: the above discussion almost proves the Fourier inversion formula. Indeed, let be the Fourier transform and the reflection; i.e., . Then, dealing directly with the integral that is involved, one can see commutes with coordinates and partial differentiations; hence, is a multiplication by a constant. This is almost a proof since one still has to compute this constant.)

A partial converse to the Taylor formula also holds; see Borel's lemma and Whitney extension theorem.

Inverse function theorem and submersion theorem

Inverse function theorem — Let be a map between open subsets in . If is continuously differentiable (or more generally ) and is bijective, there exists neighborhoods of and the inverse that is continuously differentiable (or respectively ).

A -map with the -inverse is called a -diffeomorphism. Thus, the theorem says that, for a map satisfying the hypothesis at a point , is a diffeomorphism near For a proof, see Inverse function theorem#A proof using successive approximation.

The implicit function theorem says:[9] given a map , if , is in a neighborhood of and the derivative of at is invertible, then there exists a differentiable map for some neighborhoods of such that . The theorem follows from the inverse function theorem; see Inverse function theorem#Implicit function theorem.

Another consequence is the submersion theorem.

Integrable functions on Euclidean spaces

A partition of an interval is a finite sequence . A partition of a rectangle (product of intervals) in then consists of partitions of the sides of ; i.e., if , then consists of such that is a partition of .[10]

Given a function on , we then define the upper Riemann sum of it as:

where

The lower Riemann sum of is then defined by replacing by . Finally, the function is called integrable if it is bounded and . In that case, the common value is denoted as .[12]

A subset of is said to have measure zero if for each , there are some possibly infinitely many rectangles whose union contains the set and [13]

A key theorem is

Theorem — [14] A bounded function on a closed rectangle is integrable if and only if the set has measure zero.

The next theorem allows us to compute the integral of a function as the iteration of the integrals of the function in one-variables:

Fubini's theorem — If is a continuous function on a closed rectangle (in fact, this assumption is too strong), then

In particular, the order of integrations can be changed.

Finally, if is a bounded open subset and a function on , then we define where is a closed rectangle containing and is the characteristic function on ; i.e., if and if provided is integrable.[15]

Surface integral

If a bounded surface in is parametrized by with domain , then the surface integral of a measurable function on is defined and denoted as:

If is vector-valued, then we define

where is an outward unit normal vector to . Since , we have:

Vector analysis

Tangent vectors and vector fields

Let be a differentiable curve. Then the tangent vector to the curve at is a vector at the point whose components are given as:

.[16]

For example, if is a helix, then the tangent vector at t is:

It corresponds to the intuition that the a point on the helix moves up in a constant speed.

If is a differentiable curve or surface, then the tangent space to at a point p is the set of all tangent vectors to the differentiable curves with .

A vector field X is an assignment to each point p in M a tangent vector to M at p such that the assignment varies smoothly.

Differential forms

The dual notion of a vector field is a differential form. Given an open subset in , by definition, a differential 1-form (often just 1-form) is an assignment to a point in a linear functional on the tangent space to at such that the assignment varies smoothly. For a (real or complex-valued) smooth function , define the 1-form by: for a tangent vector at ,

where denotes the directional derivative of in the direction at .[17] For example, if is the -th coordinate function, then ; i.e., are the dual basis to the standard basis on . Then every differential 1-form can be written uniquely as

for some smooth functions on (since, for every point , the linear functional is a unique linear combination of over real numbers). More generally, a differential k-form is an assignment to a point in a vector in the -th exterior power of the dual space of such that the assignment varies smoothly.[17] In particular, a 0-form is the same as a smooth function. Also, any -form can be written uniquely as:

for some smooth functions .[17]

Like a smooth function, we can differentiate and integrate differential forms. If is a smooth function, then can be written as:[18]

since, for , we have: . Note that, in the above expression, the left-hand side (whence the right-hand side) is independent of coordinates ; this property is called the invariance of differential.

The operation is called the exterior derivative and it extends to any differential forms inductively by the requirement (Leibniz rule)

where are a p-form and a q-form.

The exterior derivative has the important property that ; that is, the exterior derivative of a differential form is zero. This property is a consequence of the symmetry of second derivatives (mixed partials are equal).

Boundary and orientation

A circle can be oriented clockwise or counterclockwise. Mathematically, we say that a subset of is oriented if there is a consistent choice of normal vectors to that varies continuously. For example, a circle or, more generally, an n-sphere can be oriented; i.e., orientable. On the other hand, a Möbius strip (a surface obtained by identified by two opposite sides of the rectangle in a twisted way) cannot oriented: if we start with a normal vector and travel around the strip, the normal vector at end will point to the opposite direction.

Proposition — A bounded differentiable region in of dimension is oriented if and only if there exists a nowhere-vanishing -form on (called a volume form).

The proposition is useful because it allows us to give an orientation by giving a volume form.

Integration of differential forms

If is a differential n-form on an open subset M in (any n-form is that form), then the integration of it over with the standard orientation is defined as:

If M is given the orientation opposite to the standard one, then is defined as the negative of the right-hand side.

Then we have the fundamental formula relating exterior derivative and integration:

Stokes' formula — For a bounded region in of dimension whose boundary is a union of finitely many -subsets, if is oriented, then

for any differential -form on the boundary of .

Here is a sketch of proof of the formula.[19] If is a smooth function on with compact support, then we have:

(since, by the fundamental theorem of calculus, the above can be evaluated on boundaries of the set containing the support.) On the other hand,

Let approach the characteristic function on . Then the second term on the right goes to while the first goes to , by the argument similar to proving the fundamental theorem of calculus.

The formula generalizes the fundamental theorem of calculus as well as Stokes' theorem in multivariable calculus. Indeed, if is an interval and , then and the formula says:

.

Similarly, if is an oriented bounded surface in and , then and similarly for and . Collecting the terms, we thus get:

Then, from the definition of the integration of , we have where is the vector-valued function and . Hence, Stokes’ formula becomes

which is the usual form of the Stokes' theorem on surfaces. Green’s theorem is also a special case of Stokes’ formula.

Stokes' formula also yields a general version of Cauchy's integral formula. To state and prove it, for the complex variable and the conjugate , let us introduce the operators

In these notations, a function is holomorphic (complex-analytic) if and only if (the Cauchy–Riemann equations). Also, we have:

Let be a punctured disk with center . Since is holomorphic on , We have:

.

By Stokes’ formula,

Letting we then get:[20][21]

Winding numbers and Poincaré lemma

This section needs expansion. You can help by adding to it. (May 2022)

A differential form is called closed if and is called exact if for some differential form (often called a potential). Since , an exact form is closed. But the converse does not hold in general; there might be a non-exact closed form. A classic example of such a form is:[22]

,

which is a differential form on . Suppose we switch to polar coordinates: where . Then

This does not show that is exact: the trouble is that is not a well-defined continuous function on . Since any function on with differ from by constant, this means that is not exact. The calculation, however, shows that is exact, for example, on since we can take there.

There is a result (Poincaré lemma) that gives a condition that guarantees closed forms are exact. To state it, we need some notions from topology. Given two continuous maps between subsets of (or more generally topological spaces), a homotopy from to is a continuous function such that and . Intuitively, a homotopy is a continuous variation of one function to another. A loop in a set is a curve whose starting point coincides with the end point; i.e., such that . Then a subset of is called simply connected if every loop is homotopic to a constant function. A typical example of a simply connected set is a disk . Indeed, given a loop , we have the homotopy from to the constant function . A punctured disk, on the other hand, is not simply connected.

Poincaré lemma — If is a simply connected open subset of , then each closed form on is exact.

Geometry of curves and surfaces

This section needs expansion. You can help by adding to it. (May 2022)

Moving frame

Vector fields on are called a frame field if they are orthogonal to each other at each point; i.e., at each point.[23] The basic example is the standard frame ; i.e., is a standard basis for each point in . Another example is the cylindrical frame

[24]

For the study of the geometry of a curve, the important frame to use is a Frenet frame on a unit-speed curve given as:

The Gauss–Bonnet theorem

The Gauss–Bonnet theorem relates the topology of a surface and its geometry.

The Gauss–Bonnet theorem — [25] For each bounded surface in , we have:

where is the Euler characteristic of and the curvature.

Calculus of variations

Method of Lagrange multiplier

Lagrange multiplier — [26] Let be a differentiable function from an open subset of such that has rank at every point in . For a differentiable function , if attains either a maximum or minimum at a point in , then there exists real numbers such that

.

In other words, is a stationary point of .

The set is usually called a constraint.

Example:[27] Suppose we want to find the minimum distance between the circle and the line . That means that we want to minimize the function , the square distance between a point on the circle and a point on the line, under the constraint . We have:

Since the Jacobian matrix of has rank 2 everywhere on , the Lagrange multiplier gives:

If , then , not possible. Thus, and

From this, it easily follows that and . Hence, the minimum distance is (as a minimum distance clearly exists).

Here is an application to linear algebra.[28] Let be a finite-dimensional real vector space and a self-adjoint operator. We shall show has a basis consisting of eigenvectors of (i.e., is diagonalizable) by induction on the dimension of . Choosing a basis on we can identify and is represented by the matrix . Consider the function , where the bracket means the inner product. Then . On the other hand, for , since is compact, attains a maximum or minimum at a point in . Since , by Lagrange multiplier, we find a real number such that But that means . By inductive hypothesis, the self-adjoint operator , the orthogonal complement to , has a basis consisting of eigenvectors. Hence, we are done. .

Weak derivatives

Up to measure-zero sets, two functions can be determined to be equal or not by means of integration against other functions (called test functions). Namely, the following sometimes called the fundamental lemma of calculus of variations:

Lemma[29] — If are locally integrable functions on an open subset such that

for every (called a test function). Then almost everywhere. If, in addition, are continuous, then .

Given a continuous function , by the lemma, a continuously differentiable function is such that if and only if

for every . But, by integration by parts, the partial derivative on the left-hand side of can be moved to that of ; i.e.,

where there is no boundary term since has compact support. Now the key point is that this expression makes sense even if is not necessarily differentiable and thus can be used to give sense to a derivative of such a function.

Note each locally integrable function defines the linear functional on and, moreover, each locally integrable function can be identified with such linear functional, because of the early lemma. Hence, quite generally, if is a linear functional on , then we define to be the linear functional where the bracket means . It is then called the weak derivative of with respect to . If is continuously differentiable, then the weak derivate of it coincides with the usual one; i.e., the linear functional is the same as the linear functional determined by the usual partial derivative of with respect to . A usual derivative is often then called a classical derivative. When a linear functional on is continuous with respect to a certain topology on , such a linear functional is called a distribution, an example of a generalized function.

A classic example of a weak derivative is that of the Heaviside function , the characteristic function on the interval .[30] For every test function , we have:

Let denote the linear functional , called the Dirac delta function (although not exactly a function). Then the above can be written as:

Cauchy's integral formula has a similar interpretation in terms of weak derivatives. For the complex variable , let . For a test function , if the disk contains the support of , by Cauchy's integral formula, we have:

Since , this means:

or

[31]

In general, a generalized function is called a fundamental solution for a linear partial differential operator if the application of the operator to it is the Dirac delta. Hence, the above says is the fundamental solution for the differential operator .

See also: Limit of distributions

Hamilton–Jacobi theory

Main article: Hamilton–Jacobi equation

This section needs expansion. You can help by adding to it. (May 2022)

Calculus on manifolds

Definition of a manifold

This section requires some background in general topology.

A manifold is a Hausdorff topological space that is locally modeled by an Euclidean space. By definition, an atlas of a topological space is a set of maps , called charts, such that

By definition, a manifold is a second-countable Hausdorff topological space with a maximal atlas (called a differentiable structure); "maximal" means that it is not contained in strictly larger atlas. The dimension of the manifold is the dimension of the model Euclidean space ; namely, and a manifold is called an n-manifold when it has dimension n. A function on a manifold is said to be smooth if is smooth on for each chart in the differentiable structure.

A manifold is paracompact; this has an implication that it admits a partition of unity subordinate to a given open cover.

If is replaced by an upper half-space , then we get the notion of a manifold-with-boundary. The set of points that map to the boundary of under charts is denoted by and is called the boundary of . This boundary may not be the topological boundary of . Since the interior of is diffeomorphic to , a manifold is a manifold-with-boundary with empty boundary.

The next theorem furnishes many examples of manifolds.

Theorem — [32] Let be a differentiable map from an open subset such that has rank for every point in . Then the zero set is an -manifold.

For example, for , the derivative has rank one at every point in . Hence, the n-sphere is an n-manifold.

The theorem is proved as a corollary of the inverse function theorem.

Many familiar manifolds are subsets of . The next theoretically important result says that there is no other kind of manifolds. An immersion is a smooth map whose differential is injective. An embedding is an immersion that is homeomorphic (thus diffeomorphic) to the image.

Whitney's embedding theorem — Each -manifold can be embedded into .

The proof that a manifold can be embedded into for some N is considerably easier and can be readily given here. It is known[citation needed] that a manifold has a finite atlas . Let be smooth functions such that and cover (e.g., a partition of unity). Consider the map

It is easy to see that is an injective immersion. It may not be an embedding. To fix that, we shall use:

where is a smooth proper map. The existence of a smooth proper map is a consequence of a partition of unity. See [1] for the rest of the proof in the case of an immersion.

Nash's embedding theorem says that, if is equipped with a Riemannian metric, then the embedding can be taken to be isometric with an expense of increasing ; for this, see this T. Tao's blog.

Tubular neighborhood and transversality

This section needs expansion. You can help by adding to it. (May 2022)

A technically important result is:

Tubular neighborhood theorem — Let M be a manifold and a compact closed submanifold. Then there exists a neighborhood of such that is diffeomorphic to the normal bundle to and corresponds to the zero section of under the diffeomorphism.

This can be proved by putting a Riemannian metric on the manifold . Indeed, the choice of metric makes the normal bundle a complementary bundle to ; i.e., is the direct sum of and . Then, using the metric, we have the exponential map for some neighborhood of in the normal bundle to some neighborhood of in . The exponential map here may not be injective but it is possible to make it injective (thus diffeomorphic) by shrinking (for now, see see [2]).


Integration on manifolds and distribution densities

This section needs expansion. You can help by adding to it. (May 2022)

The starting point for the topic of integration on manifolds is that there is no invariant way to integrate functions on manifolds. This may be obvious if we asked: what is an integration of functions on a finite-dimensional real vector space? (In contrast, there is an invariant way to do differentiation since, by definition, a manifold comes with a differentiable structure). There are several ways to introduce integration theory to manifolds:

For example, if a manifold is embedded into an Euclidean space , then it acquires the Lebesgue measure restricting from the ambient Euclidean space and then the second approach works. The first approach is fine in many situations but it requires the manifold to be oriented (and there is an non-orientable manifold that is not pathological). The third approach generalizes and that gives rise to the notion of a density.

Generalizations

Extensions to infinite-dimensional normed spaces

The notions like differentiability extend to normed spaces.

See also

Notes

  1. ^ This is just the tensor-hom adjunction.

Citations

  1. ^ Spivak 1965, Ch 2. Basic definitions.
  2. ^ Hörmander 2015, Definition 1.1.4.
  3. ^ Hörmander 2015, (1.1.3.)
  4. ^ Hörmander 2015, Theorem 1.1.6.
  5. ^ Hörmander 2015, (1.1.2)'
  6. ^ Hörmander 2015, p. 8
  7. ^ a b Hörmander 2015, Theorem 1.1.8.
  8. ^ Hörmander 2015, Lemma 7.1.4.
  9. ^ Spivak 1965, Theorem 2-12.
  10. ^ Spivak 1965, p. 46
  11. ^ Spivak 1965, p. 47
  12. ^ Spivak 1965, p. 48
  13. ^ Spivak 1965, p. 50
  14. ^ Spivak 1965, Theorem 3-8.
  15. ^ Spivak 1965, p. 55
  16. ^ Spivak 1965, Exercise 4.14.
  17. ^ a b c Spivak 1965, p. 89
  18. ^ Spivak 1965, Theorem 4-7.
  19. ^ Hörmander 2015, p. 151
  20. ^ Theorem 1.2.1. in Hörmander, Lars (1990). An Introduction to Complex Analysis in Several Variables (Third ed.). North Holland..
  21. ^ Spivak 1965, Exercise 4-33.
  22. ^ Spivak 1965, p. 93
  23. ^ O'Neill 2006, Definition 6.1.
  24. ^ O'Neill 2006, Example 6.2. (1)
  25. ^ O'Neill 2006, Theorem 6.10.
  26. ^ Spivak 1965, Exercise 5-16.
  27. ^ Edwards 1994, Ch. II, $ 5. Example 9.
  28. ^ Spivak 1965, Exercise 5-17.
  29. ^ Hörmander 2015, Theorem 1.2.5.
  30. ^ Hörmander 2015, Example 3.1.2.
  31. ^ Hörmander 2015, p. 63
  32. ^ Spivak 1965, Theorem 5-1.

References