Search results
Results from the WOW.Com Content Network
The divergence of a vector field extends naturally to any differentiable manifold of dimension n that has a volume form (or density) μ, e.g. a Riemannian or Lorentzian manifold. Generalising the construction of a two-form for a vector field on R 3, on such a manifold a vector field X defines an (n − 1)-form j = i X μ obtained by contracting ...
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
In 3 dimensions, a differential 0-form is a real-valued function (,,); a differential 1-form is the following expression, where the coefficients are functions: + +; a differential 2-form is the formal sum, again with function coefficients: + +; and a differential 3-form is defined by a single term with one function as coefficient: .
When this happens, the limit of the product of these two factors will equal the product of the limits of the factors. The two factors are Q(g(x)) and (g(x) − g(a)) / (x − a). The latter is the difference quotient for g at a, and because g is differentiable at a by assumption, its limit as x tends to a exists and equals g′(a).
The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides. If the matrix entries are real numbers, the matrix A can be used to represent two linear maps: one that maps the standard basis vectors to the rows of A, and one that maps them to the columns of A.
Even if the gradient theorem (also called fundamental theorem of calculus for line integrals) has been proved for a differentiable (so looked as smooth) curve so far, the theorem is also proved for a piecewise-smooth curve since this curve is made by joining multiple differentiable curves so the proof for this curve is made by the proof per ...
There are two lists of mathematical identities related to vectors: Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
This is because the n-dimensional dV element is in general a parallelepiped in the new coordinate system, and the n-volume of a parallelepiped is the determinant of its edge vectors. The Jacobian can also be used to determine the stability of equilibria for systems of differential equations by approximating behavior near an equilibrium point.