Dirichlet-Jordan Theorem

Inverse Fourier Transform

Dominated Convergence Theorem

Smooth Function & Analytic Function

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Exterior Derivatives of Differential Forms

Write out first-order form L as:

and replace each of the scalar factors (A, B, C) by its differential, formed in the ordinary sense.

Thus, if we associate with L the vector R=(A,B,C), we have the remarkable fact that the coefficients of dL are just the components of the curl of R.

The exterior derivative extends the concept of the differential of a function to differential forms of higher degree. (function itself is differential form of 0 degree)

Closed Differential Form & Exact Differential Form

If the coefficients of the differential form

has continuous first derivatives in a simply connected set R and satisfy the integrability conditions

then L is the total differential of a function u defined in R, and: