It is also conceptually helpful to do this in an invariant manner (i.e., a coordinate-free way).
be the set of all invertible real square matrices of size n. Note
stands for homomorphisms between vector spaces; i.e., linear maps.
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.
A partial converse to the Taylor formula also holds; see Borel's lemma and Whitney extension theorem.
at a point p is the set of all tangent vectors to the differentiable curves
The dual notion of a vector field is a differential form.
is called the exterior derivative and it extends to any differential forms inductively by the requirement (Leibniz rule) where
This property is a consequence of the symmetry of second derivatives (mixed partials are equal).
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.
Green’s theorem is also a special case of Stokes’ formula.
There is a result (Poincaré lemma) that gives a condition that guarantees closed forms are exact.
is called simply connected if every loop is homotopic to a constant function.
A typical example of a simply connected set is a disk
Example:[27] Suppose we want to find the minimum distance between the circle
But, by integration by parts, the partial derivative on the left-hand side of
is not necessarily differentiable and thus can be used to give sense to a derivative of such a function.
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
A classic example of a weak derivative is that of the Heaviside function
Then the above can be written as: Cauchy's integral formula has a similar interpretation in terms of weak derivatives.
, this means: or 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.
, 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.
A manifold is paracompact; this has an implication that it admits a partition of unity subordinate to a given open cover.
The existence of a smooth proper map is a consequence of a partition of unity.
is equipped with a Riemannian metric, then the embedding can be taken to be isometric with an expense of increasing
A technically important result is: Tubular neighborhood theorem — Let M be a manifold and
This may be obvious if we asked: what is an integration of functions on a finite-dimensional real vector space?
, then it acquires the Lebesgue measure restricting from the ambient Euclidean space and then the second approach works.