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What is the exterior derivative of a function?

The exterior derivative is defined to be the unique ℝ -linear mapping from k -forms to (k + 1) -forms satisfying the following properties: df is the differential of f , for 0 -forms ( smooth functions) f . d(df ) = 0 for any 0 -form (smooth function) f .

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 .

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 product of differential forms?

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 . The exterior product of differential forms (denoted with the same symbol ∧) is defined as their pointwise exterior product .


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