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  2. Gradient - Wikipedia

    en.wikipedia.org/wiki/Gradient

    The gradient of the function f(x,y) = −(cos 2 x + cos 2 y) 2 depicted as a projected vector field on the bottom plane. The gradient (or gradient vector field) of a scalar function f(x 1, x 2, x 3, …, x n) is denoted ∇f or ∇ → f where ∇ denotes the vector differential operator, del.

  3. Vector calculus identities - Wikipedia

    en.wikipedia.org/wiki/Vector_calculus_identities

    More generally, for a function of n variables (, …,), also called a scalar field, the gradient is the vector field: = (, …,) = + + where (=,,...,) are mutually orthogonal unit vectors. As the name implies, the gradient is proportional to, and points in the direction of, the function's most rapid (positive) change.

  4. Gradient theorem - Wikipedia

    en.wikipedia.org/wiki/Gradient_theorem

    The gradient theorem states that if the vector field F is the gradient of some scalar-valued function (i.e., if F is conservative), then F is a path-independent vector field (i.e., the integral of F over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse:

  5. Vector field - Wikipedia

    en.wikipedia.org/wiki/Vector_field

    A vector field V defined on an open set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that = = (,,, …,). The associated flow is called the gradient flow , and is used in the method of gradient descent .

  6. Conservative vector field - Wikipedia

    en.wikipedia.org/wiki/Conservative_vector_field

    In vector calculus, a conservative vector field is a vector field that is the gradient of some function. [1] A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the ...

  7. Matrix calculus - Wikipedia

    en.wikipedia.org/wiki/Matrix_calculus

    By example, in physics, the electric field is the negative vector gradient of the electric potential. The directional derivative of a scalar function f(x) of the space vector x in the direction of the unit vector u (represented in this case as a column vector) is defined using the gradient as follows.

  8. Helmholtz decomposition - Wikipedia

    en.wikipedia.org/wiki/Helmholtz_decomposition

    The Helmholtz decomposition in three dimensions was first described in 1849 [9] by George Gabriel Stokes for a theory of diffraction. Hermann von Helmholtz published his paper on some hydrodynamic basic equations in 1858, [10] [11] which was part of his research on the Helmholtz's theorems describing the motion of fluid in the vicinity of vortex lines. [11]

  9. Tensor derivative (continuum mechanics) - Wikipedia

    en.wikipedia.org/wiki/Tensor_derivative...

    Consider a vector field v and an arbitrary constant vector c. In index notation, the cross product is given by v × c = ε i j k v j c k e i {\displaystyle \mathbf {v} \times \mathbf {c} =\varepsilon _{ijk}~v_{j}~c_{k}~\mathbf {e} _{i}} where ε i j k {\displaystyle \varepsilon _{ijk}} is the permutation symbol , otherwise known as the Levi ...