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with the derivative evaluated at = Another connexion with the confluent hypergeometric functions is that E 1 is an exponential times the function U(1,1,z): = (,,) The exponential integral is closely related to the logarithmic integral function li(x) by the formula
In the theory of differential forms, a differential ideal I is an algebraic ideal in the ring of smooth differential forms on a smooth manifold, in other words a graded ideal in the sense of ring theory, that is further closed under exterior differentiation d, meaning that for any form α in I, the exterior derivative dα is also in I.
In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras are used for the study of algebraic varieties, which are solution sets of ...
The following table lists the number of smooth types of the topological m−sphere S m for the values of the dimension m from 1 up to 20. Spheres with a smooth, i.e. C ∞ −differential structure not smoothly diffeomorphic to the usual one are known as exotic spheres.
Every solution of the second half g of the equation defines a unique direction for x via the first half f of the equations, while the direction for y is arbitrary. But not every point (x,y,t) is a solution of g. The variables in x and the first half f of the equations get the attribute differential.
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In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold.
Using the chain rule, the reciprocal rule, and the fact that the derivative of the exponential function is again the exponential function, we see that the formula is correct for the first derivative of f for all x > 0 and that p 1 (x) is a polynomial of degree 0. Of course, the derivative of f is zero for x < 0.