Gradient
Gradient of a vector field

Since the total derivative of a vector field is a linear mapping from vectors to vectors, it is a tensor quantity.

In rectangular coordinates, the gradient of a vector field f = (f, f, f) is defined by:

∇ f = g j k ∂ f i ∂ x j e i ⊗ e k, {\displaystyle \nabla \mathbf {f} =g^{jk}{\frac {\partial f^{i}}{\partial x^{j}}}\mathbf {e} _{i}\otimes \mathbf {e} _{k},}

(where the Einstein summation notation is used and the tensor product of the vectors e and e is a dyadic tensor of type (2,0)). Overall, this expression equals the transpose of the Jacobian matrix:

∂ f i ∂ x j = ∂ (f 1, f 2, f 3) ∂ (x 1, x 2, x 3). {\displaystyle {\frac {\partial f^{i}}{\partial x^{j}}}={\frac {\partial (f^{1},f^{2},f^{3})}{\partial (x^{1},x^{2},x^{3})}}.}

In curvilinear coordinates, or more generally on a curved manifold, the gradient involves Christoffel symbols:

∇ f = g j k (∂ f i ∂ x j + Γ i j l f l) e i ⊗ e k, {\displaystyle \nabla \mathbf {f} =g^{jk}\left({\frac {\partial f^{i}}{\partial x^{j}}}+{\Gamma ^{i}}_{jl}f^{l}\right)\mathbf {e} _{i}\otimes \mathbf {e} _{k},}

where g are the components of the inverse metric tensor and the e are the coordinate basis vectors.

Expressed more invariantly, the gradient of a vector field f can be defined by the Levi-Civita connection and metric tensor:

∇ a f b = g a c ∇ c f b, {\displaystyle \nabla ^{a}f^{b}=g^{ac}\nabla _{c}f^{b},}

where ∇ is the connection.
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