Vector calculus identities

The following are important identities involving derivatives and integrals in vector calculus. == Operator notation == === Gradient === For a function f ( x , y , z ) {\displaystyle f(x,y,z)} in three-dimensional Cartesian coordinate variables, the gradient is the vector field: grad ⁡ ( f ) = ∇ f = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) f = ∂ f ∂ x i + ∂ f ∂ y j + ∂ f ∂ z k {\displaystyle \operatorname {grad} (f)=\nabla f={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} } where i, j, k are the standard unit vectors for the x, y, z-axes.

Source: Wikipedia — Vector calculus identities (CC BY-SA 4.0)

Vector calculus identities

The following are important identities involving derivatives and integrals in vector calculus. == Operator notation == === Gradient === For a function f ( x , y , z ) {\displaystyle f(x,y,z)} in three-dimensional Cartesian coordinate variables, the gradient is the vector field: grad ⁡ ( f ) = ∇ f = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) f = ∂ f ∂ x i + ∂ f ∂ y j + ∂ f ∂ z k {\displaystyle \operatorname {grad} (f)=\nabla f={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} } where i, j, k are the standard unit vectors for the x, y, z-axes.

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Source: Wikipedia "Vector calculus identities" · CC BY-SA 4.0

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