GATE MATHEMATICS : CALCULUS

ONLINE TEST : GATE MATHS :- CALCULUS

Chain Rule :

Let w = f(x,y,z) be continuous and have continuous first partial derivative in a domain D in xyz-space. Let x = x(u,v), y = y(u,v), z = z(u,v) be functions that are continuous and have first partial derivatives in a domain B in the uv-plane, where B is such that for every point (u,v) in B, the corresponding point [x(u,v), y(u,v), z(u,v)] lies in D. Then the function

w = f(x(u,v), y(u,v), z(u,v))

is defined in B, has partial derivatives w.r.t. u and v in B,

$\bg_white \frac{\partial w}{\partial u} = \frac{\partial w}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial w}{\partial y}\frac{\partial y}{\partial u} + \frac{\partial w}{\partial z}\frac{\partial z}{\partial u}$

$\bg_white \frac{\partial w}{\partial v} = \frac{\partial w}{\partial x}\frac{\partial x}{\partial v} + \frac{\partial w}{\partial y}\frac{\partial y}{\partial v} + \frac{\partial w}{\partial z}\frac{\partial z}{\partial v}$

Mean Value Theorem :

Let f(x,y,z) be continuous and have continuous first partial derivative in a domain D in xyz space. Let P₀ :(x₀,y₀,z₀) and P : (x₀ + h,y₀ + k,z₀ + l) be points in D such that the straight line segment P₀P joining these points lies entirely in D. Then

$\bg_white f(x_0 + h, y_0 + k, z_0 + l) - f(x_0,y_0,z_0) = h\frac{\partial f}{\partial x} + l\frac{\partial f}{\partial y} + l\frac{\partial f}{\partial z}$

Indefinite Integrals :

If f(x) and F(x) are two functions of x such that
d/dx{f(x)} = F(x), then integral of F(x) is

$\bg_white \int F(x) dx = f(x) + c$
where c is any constant. Due to the addition of 'c' it is an indefinite integral.

Definite Integrals

If f(x) is a continuous functions of x in [a,b] and ψ(x) is another function of x such that

$\bg_white \frac{d}{dx}{[\psi (x)]} = f(x)$
then
$\bg_white \int_{a}^{b}f(x)dx = \psi (b) - \psi (a)$

Theorem of Integral Calculus :

$\bg_white \int_{a}^{b}f(x)dx = \lim_{h\rightarrow 0} h[f(a) + f(a + h) + f(a + 2h) + . . . + f(a + (n - 1)h]\\\\ where n = \infty, h = 0, nh = b - a$

Gamma Function :

$\bg_white \Gamma{n} = \int_{0}^{\infty} e^{-x} x^{n-1}dx,\\\\ where\\ n\ is\ a\ positive\ integer.$

Beta Function :

$\bg_white \Gamma{n} = \int_{0}^{\infty} e^{-x} x^{n-1}dx,\\\\ where\\ n\ is\ a\ positive\ integer.$

Euler's Theorem on Homogeneous Function:

If z is homogeneous function of x, y of order n, then

Taylor's Series of Two Variables :

If f(x, y) and all its partial derivatives up to nth order are finite and continuous for all points (s, y),
where a ≤ x ≤ a + h, a ≤ y ≤ a + k
Then

$\bg_white f(a+h, b+k ) = f(a,b) + (h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y})f\\ + \frac{1}{2!}(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y})^{2}f + \frac{1}{3!}(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y})^{3}f = ...$

Maximum Value Theorem:

A function of x,y will be the maximum at x = a, y = b.

if f(a, b) > f(a+h, b+k)
or
if f(a+h, b+k) - f(a,b) < 0, then f(a,b) is maximum.

Minimum Value Theorem

A function of x,y will be the minimum at x = a, y = b.

if f(a, b) < f(a+h, b+k) or if f(a+h, b+k) - f(a,b) > 0, then f(a,b) is minimum.

Line Integral :

If $\bg_white \overrightarrow F (x, y, z)$ is a vector function.
Then Line Integral of $\bg_white \overrightarrow F$ along any given curve DC will be
$\bg_white \int_{c}\left [F . \frac{d\overrightarrow r}{ds} \right ]ds$

where $\bg_white \int_{c}^{}\left [F . \frac{d\overrightarrow r}{ds} \right ]ds$ is a unit vector along tangent of curve PQ

$\bg_white \int_{c} \left [F . \frac{d\overrightarrow r}{ds} \right ]ds$

Surface Integral :

Surface integral of $\bg_white \overrightarrow F$ over S is given surface

$\iint_{s} \left (\overrightarrow F . \hat n \right) ds$
where
$\bg_white \hat {n} = \frac{grad\ f}{|grad\ f|}.dS = \frac{dx . dy}{\left((\hat n . k)\right)}$

Volume Integral :

Volume integral of over the volume V enclosed by a closed surface.

$\bg_white \ =\ \int \iint_{V} \overrightarrow F . dV$

Green's Theorem :

$\bg_white \oint _{C} (\phi dx + \psi dy) = \iint_{R}\left (\frac{\partial \psi}{\partial x} - \frac{\partial \phi}{\partial y}\right )dxdy$
where
$\bg_white \phi (x,y), \psi (x,y), \frac{\partial \phi}{\partial y},\frac{\partial \psi}{\partial x}$
are continuous function over a region R bounded by simple closed curve C in x-y plane.

Stroke's Theorem :

$\bg_white \oint \overrightarrow F . d\overrightarrow r = \iint_{S}\left (\bigtriangledown X \overrightarrow F\right ).\overrightarrow n dS \\\\ =\iint_{S}\left (\bigtriangledown X \overrightarrow F\right ).dS \\\\\\ where\\ \hat n = cos \alpha\ \hat i + cos \beta\ \hat j + + cos \gamma\ \hat k\\$

is unit external normal to any surface dS.

Gauss's Theorem of Divergence :

The surface integral of the normal component of a vector function $\bg_white \overrightarrow F$ taken around a closed surface S is equal to the integral of the divergence of $\bg_white \overrightarrow F$ taken over the volume V enclosed by the surface S.
$\bg_white \iint_{S}\overrightarrow F.\hat n .dS = \int \iint_{V} div \overrightarroe F dV$

QUIZLE - Who Wants To Be An Quizmaster ?

Wednesday, October 14, 2009