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Search Results for "derivative" Documents 1 - 10 of 225 matches. 1 2345678910[next] 
Derivative--from Eric Weisstein's World of Mathematics Derivative The derivative of a function represents an infinitesimal change in the function with respect to whatever parameters it may have. The ``simple'' derivative of a function $f$ with respect to $x$ is denoted either $f'(x)$ or \begin{displaymath} {df\over dx} \end{displaymath} (1) (and often... http://mathworld.wolfram.com/Derivative.html, 33507 bytes 
Covariant Derivative--from Eric Weisstein's World of Mathematics Covariant Derivative The covariant derivative of a contravariant tensor $A^a$ (also called the ``semicolon derivative'' since its symbol is a semicolon) is given by \begin{displaymath} \nabla \cdot\mathbf{A} \equiv {A^a}_{;b} = {A^a}_{,b}+\Gamma^a_{bk}\,A^k, \end{displaymath} (1) where $A^k_{,k}$ is... http://mathworld.wolfram.com/CovariantDerivative.html, 14605 bytes
Second Derivative Test--from Eric Weisstein's World of Mathematics Second Derivative Test Suppose $f(x)$ is a function of $x$ which is twice differentiable at a stationary point $x_0$ . 1. If $f''(x_0) > 0$ , then $f$ has a relative minimum at $x_0$ . 2. If $f''(x_0) < 0$ , then $f$ has a relative maximum at $x_0$ . The extremum test gives slightly more... http://mathworld.wolfram.com/SecondDerivativeTest.html, 16721 bytes
Exterior Derivative--from Eric Weisstein's World of Mathematics Exterior Derivative Consider a Differential k-Form \begin{displaymath} \omega^1 = b_1\,dx_1+b_2\,dx_2. \end{displaymath} (1) Then its exterior derivative is \begin{displaymath} d\omega^1 = db_1\wedge dx_1+db_2\wedge dx_2, \end{displaymath} (2) where $\wedge$ is the wedge product. Similarly, consider... http://mathworld.wolfram.com/ExteriorDerivative.html, 17847 bytes
Partial Derivative--from Eric Weisstein's World of Mathematics Partial Derivative Partial derivatives are defined as derivatives of a function of multiple variables when all but the variable of interest are held fixed during the differentiation. \begin{displaymath} {\partial f\over \partial x_m} \equiv\lim_{h\to 0} {f(x_1, \... ... x_m+h, \ldots, x_n)-f(x_1, ... http://mathworld.wolfram.com/PartialDerivative.html, 19303 bytes
First Derivative Test--from Eric Weisstein's World of Mathematics First Derivative Test StationaryPoint Suppose $f(x)$ is continuous at a stationary point $x_0$ . 1. If $f'(x) > 0$ on an open interval extending left from $x_0$ and $f'(x) < 0$ on an open interval extending right from $x_0$ , then $f(x)$ has a relative maximum (possibly a global maximum... http://mathworld.wolfram.com/FirstDerivativeTest.html, 15351 bytes
Directional Derivative--from Eric Weisstein's World of Mathematics Directional Derivative \begin{displaymath} \nabla_{\mathbf{u}}f \equiv\nabla f\cdot {\mathbf{u}\over\ve... ...lim_{h\to 0} {f(\mathbf{x}+h\mathbf{u})-f(\mathbf{x})\over h}. \end{displaymath} (1) $\nabla_{\mathbf{u}} f(x_0,y_0,z_0)$ is the rate at which the function $w = f(x,y,z)$ changes at $(x_0,y... http://mathworld.wolfram.com/DirectionalDerivative.html, 14441 bytes
Comma Derivative--from Eric Weisstein's World of Mathematics Comma Derivative $\displaystyle A_{,k}$ $\textstyle \equiv$ $\displaystyle {\partial A\over\partial x^k} \equiv \partial_k A$ $\displaystyle A^k_{,k}$ $\textstyle \equiv$ $\displaystyle {1\over g_k} {\partial A^k\over \partial x^k} \equiv\partial_k A^k.$ Schmutzer (1968, p. 70) uses the older notation... http://mathworld.wolfram.com/CommaDerivative.html, 13334 bytes
Logarithmic Derivative--from Eric Weisstein's World of Mathematics Logarithmic Derivative The logarithmic derivative of a function $f$ is defined as the derivative of the logarithm of a function. For example, the digamma function is defined as the logarithmic derivative of the gamma function, \begin{displaymath} \Psi(z)={d\over dz}\ln\Gamma(z). \end{displaymath} see... http://mathworld.wolfram.com/LogarithmicDerivative.html, 12822 bytes
Lie Derivative--from Eric Weisstein's World of Mathematics Lie Derivative The Lie derivative of tensor $T_{ab}$ with respect to the vector field $X$ is defined by \begin{displaymath} \mathcal{L}_X T_{ab} \equiv \lim_{\delta x\to 0} {T'_{ab}(x')-T_{ab}(x)\over\delta x}. \end{displaymath} (1) Explicitly, it is given by \begin{displaymath} \mathcal{L}_X T_{ab... http://mathworld.wolfram.com/LieDerivative.html, 14292 bytes 1 2345678910[next] |
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