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The differential of a function f at x0 is simply the linear function which produces the best linear approximation of f(x) in a neighbourhood of x0. Specifically, among the linear functions l that take the value f(x0) at x0, there exists at most one such that, in a neighbourhood of x0, we have: f(x0 + h) = f(x0) + l(h) + o(h) It is the linear ...
In simple words, the rate of change of function is called as a derivative and differential is the actual change of function. We can also define a derivative in terms of differentials as the ratio of differentials of function by the differential of a variable.
We are finally ready to define a differential form. A differential k -form on an n dimensional smooth manifold M is any multilinear function ω which takes as input k smooth vector fields on M, X1, ⋯, Xk and outputs a scalar function on M so that ω(X1, ⋯, Xi, ⋯, Xj, ⋯, Xk) = − ω(X1, ⋯, Xj, ⋯, Xi, ⋯, Xk). The latter property is ...
$\begingroup$ Similar to what you wrote about "algebra", the statement "calculus, discrete mathematics, and geometry, are independent enough" really depends on what the OP meant by geometry. Classical euclidean geometry? I agree. Analytical and differential geometry? I think those are rather deeply connected with calculus. Metric geometry?
Differential calculus is a child while integral calculus is grand parent. One first learns the evolution of child and then understands the old person. Differential coefficient, being tangent of inclination of function, is akin to psychology and behavioral pattern of the child; while integral calculus approach is the tendency of old generation ...
The post does not imply that differential forms are the only formalization of differentials, only the most common (which is true), and that any formalization is too complex for first year calc students, which is even more true of NSA than it is of differential forms.
I usually think of multivariable calculus as being divided into four parts: (Partial) Differentiation. (Multiple) Integration. Curves and Surfaces in R3 R 3. Vector Calculus (Green's Theorem, Stokes' Theorem, Divergence Theorem) For differentiation, you can use Principles of Mathematical Analysis by Rudin (Chapter 9).
$\begingroup$ I would also recommend Schaum's book for calculus, and Ordinary Differential Equations by Pollard and Tenenbaum for DEs. $\endgroup$ – Chris Taylor Commented May 25, 2011 at 7:49
There is something in J. T. Schwartz's book on Nonlinear functional analysis, Gordon and Breach. The most complete source is, as far as I know, the book by Cartan, Differential calculus, Hermann. Anyway, you should keep in mind that differential calculus in normed spaces is rather easy and classical. Integration theory becomes more intriguing ...
The symbol. dy dx d y d x. means the derivative of y y with respect to x x. If y = f(x) y = f (x) is a function of x x, then the symbol is defined as. dy dx =limh→0 f(x + h) − f(x) h. d y d x = lim h → 0 f (x + h) − f (x) h. and this is is (again) called the derivative of y y or the derivative of f f. Note that it again is a function of ...