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How do you find the exterior derivative of a function?

The exterior derivative of a function is the one-form. (1) written in a coordinate chart . Thinking of a function as a zero-form, the exterior derivative extends linearly to all differential k -forms using the formula. (2) when is a -form and where is the wedge product . The exterior derivative of a -form is a -form.

What is the exterior derivative of a 0 -form?

The exterior derivative of this 0 -form is the 1 -form df . When an inner product ⟨·,·⟩ is defined, the gradient ∇f of a function f is defined as the unique vector in V such that its inner product with any element of V is the directional derivative of f along the vector, that is such that

What is the exterior derivative of a smooth vector field?

If f is a smooth function (a 0 -form), then the exterior derivative of f is the differential of f . That is, df is the unique 1 -form such that for every smooth vector field X, df (X) = dX f , where dX f is the directional derivative of f in the direction of X .

How do you find the exterior derivative of a differential k-form?

Thinking of a function as a zero-form, the exterior derivative extends linearly to all differential k -forms using the formula when is a -form and where is the wedge product . The exterior derivative of a -form is a -form. For example, for a differential k -form where is the permutation tensor . It is always the case that .


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