Search results
Results From The WOW.Com Content Network
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.}
This involves formulating discrete operators on graphs which are analogous to differential operators in calculus, such as graph Laplacians (or discrete Laplace operators) as discrete versions of the Laplacian, and using these operators to formulate differential equations, difference equations, or variational models on graphs which can be ...
A solution is to adopt the overdot notation, in which the scope of a vector derivative with an overdot is the multivector-valued function sharing the same overdot. In this case, if we define In this case, if we define
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
Divergence gives a measure of how much "source" or "sink" near a point there is. It can be used to calculate flux by divergence theorem. Curl measures how much "rotation" a vector field has near a point. The Lie derivative is the rate of change of a vector or tensor field along the flow of another vector field.
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.
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: