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This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
D: divergence, C: curl, G: gradient, L: Laplacian, CC: curl of curl. Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do ...
[citation needed] This is why the magnetic field, characterized by zero divergence, can be expressed as the curl of a magnetic vector potential. If W is a vector field with curl(W) = V, then adding any gradient vector field grad(f) to W will result in another vector field W + grad(f) such that curl(W + grad(f)) = V as well.
The gradient of a function is obtained by raising the index of the differential , whose components are given by: =; =; =, = = The divergence of a vector field with components is
In R 3, the gradient, curl, and divergence are special cases of the exterior derivative. An intuitive interpretation of the gradient is that it points "up": in other words, it points in the direction of fastest increase of the function. It can be used to calculate directional derivatives of scalar functions or normal directions. Divergence ...
The divergence of the curl of any vector field (in three dimensions) is equal to zero: ∇ ⋅ ( ∇ × F ) = 0. {\displaystyle \nabla \cdot (\nabla \times \mathbf {F} )=0.} If a vector field F with zero divergence is defined on a ball in R 3 , then there exists some vector field G on the ball with F = curl G .
Any of the operations of vector calculus including gradient, divergence, and curl. Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom . Functions with independent variables corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus ...
This form suggests that if we can find a function whose gradient is given by , then the integral is given by the difference of at the endpoints of the interval of integration. Thus the problem of studying the curves that make the integral stationary can be related to the study of the level surfaces of ψ . {\displaystyle \psi .}