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The derivative identity d/(dx)[f(x)g(x)] = lim_(h->0)(f(x+h)g(x+h)-f(x)g(x))/h (1) = (2) = lim_(h->0)[f(x+h)(g(x+h)-g(x))/h+g(x)(f(x+h)-f(x))/h] (3) = f(x)g^'(x)+g(x)f^'(x), ...

Given a smooth manifold M with an open cover U_i, a partition of unity subject to the cover U_i is a collection of smooth, nonnegative functions psi_i, such that the support ...

Two lines, vectors, planes, etc., are said to be perpendicular if they meet at a right angle. In R^n, two vectors a and b are perpendicular if their dot product a·b=0. (1) In ...

de Rham cohomology is a formal set-up for the analytic problem: If you have a differential k-form omega on a manifold M, is it the exterior derivative of another differential ...

An element of order 2 in a group (i.e., an element A of a group such that A^2=I, where I is the identity element).

The acceleration of an element of fluid, given by the convective derivative of the velocity v, (Dv)/(Dt)=(partialv)/(partialt)+v·del v, where del is the gradient operator.

For any function f:A->B (where A and B are any sets), the kernel (also called the null space) is defined by Ker(f)={x:x in Asuch thatf(x)=0}, so the kernel gives the elements ...

An orthogonal basis of vectors is a set of vectors {x_j} that satisfy x_jx_k=C_(jk)delta_(jk) and x^mux_nu=C_nu^mudelta_nu^mu, where C_(jk), C_nu^mu are constants (not ...

Two vectors u and v whose dot product is u·v=0 (i.e., the vectors are perpendicular) are said to be orthogonal. In three-space, three vectors can be mutually perpendicular.

The scalar triple product of three vectors A, B, and C is denoted [A,B,C] and defined by [A,B,C] = A·(BxC) (1) = B·(CxA) (2) = C·(AxB) (3) = det(ABC) (4) = |A_1 A_2 A_3; B_1 ...