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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 ...
The curl of the gradient of any scalar field φ is always the zero vector field = which follows from the antisymmetry in the definition of the curl, and the symmetry of second derivatives. The divergence of the curl of any vector field is equal to zero: ∇ ⋅ ( ∇ × F ) = 0. {\displaystyle \nabla \cdot (\nabla \times \mathbf {F} )=0.}
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 .
Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be formally defined as a vector operator whose components are the corresponding partial derivative operators.
Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc. This page was last edited on 12 October 2024, at 11:14 ...
The gradient of F is then normal to the hypersurface. Similarly, an affine algebraic hypersurface may be defined by an equation F(x 1, ..., x n) = 0, where F is a polynomial. The gradient of F is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non-singular point, it is a nonzero normal vector.
Gradient is a vector operator that operates on a scalar field, producing a vector field. Divergence is a vector operator that operates on a vector field, producing a scalar field. Curl is a vector operator that operates on a vector field, producing a vector field. Defined in terms of del:
This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using an extension of the product rule that ∇ ⋅ (ψ X) = ∇ψ ⋅X + ψ ∇⋅X: Let φ and ψ be scalar functions defined on some region U ⊂ R d, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable.