GATE MATHEMATICS : 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,


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

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

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
then

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](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ul626OtK8LFe-3CD0JVVSpDhurv7lV9rq9TB6Y7UQfz3A6-R4cg2txDXtCmH7-NuAr792FvrdokdP4gzVCME7yaILq6PKRWVVDVhs9H7RpCOMb5uFL34ZnRyKs8Bak4P5bSpE5zLEnsnCzhxWX0r3otnwz323cvLs_GV7gMbwvoVH7F-rP-iYLfrDte2fBwqKmYcRvuyMek4BKhyA_J_WeLQEcOKumNc7QI0ZxgeoGlOjL5GE-IW4X0e_OHOKlW92tl8bmvW5p530FdhF6H89oPqKMNIfs3ZJEO5oB4S2dEGW7qXM0_DZ71jrukpOBrAVf7emKuQQq4Y2_D44deGN2R1iNNTloMG9WIHd5NI8g5-OUE0bDCi82Dw50vVFV=s0-d)
Gamma Function :

Beta Function :

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

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
is a vector function.
Then Line Integral of
along any given curve DC will be
![\bg_white \int_{c}\left [F . \frac{d\overrightarrow r}{ds} \right ]ds](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_suvcyuQznWUI1BWqE3i3LL-No9cZRc06K1fPx4wwM1DvROX_WX5VOgum2vOlfZD3uvaYpV9KsQFdDLRUu717PmqEvkiDh223kTpob3zfVUtC5tkBQjNnZc5WRf9KC8NOUiko4YyTIDkP4Lq1E3rWGfXtbleY6pGiYzgF3-3NEdAWevuVF2gz5Ue4kIVgPN5FdxvlTLn_VBKRXjJjm8WyIAszHiGZSqjQJ2tyCgBBKMYqeo2Up3tJCTVfNLZcoFUg=s0-d)
where
is a unit vector along tangent of curve PQ

![\bg_white \int_{c} \left [F . \frac{d\overrightarrow r}{ds} \right ]ds](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uP9myEv_KMtg5fGHCRZK6k7pZ4zRrtBfewpX_g2ArPzadq1TFyM0IaQ4luAoRMzPrgXMQfAJBmFzqjmCAwGwQmym8H9TxjnGQaVS-60dIeOknj7_hNjmUM_dSzmtpyeXfcqNv_MHTAnaYUoxI2cspPLTgBJGPVeJa_ExGS1orj6vrNBI6ndSUbLdgYRNkA1iez9dpH4ow1C07Tve7z669ivuDS8x47QantMoYLeIR36wO9w-nh37FB91wWcIaomF1u1-A4WGTqNMbs_hOVOW_709zbdkT3LcfofNn5zNx2F7nK9UDg48yEMs2fEY8TbezclwQyVKAoss2cVwfKr0vbuPVyaLfToHXnFS3vTMTtmwsIrUcPhoy7QmIQKzunYkKuQzYlmJ-K5TcIderzhD0=s0-d)
Surface Integral :
Surface integral of
over S is given surface

where

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

Green's Theorem :

where

are continuous function over a region R bounded by simple closed curve C in x-y plane.
Stroke's Theorem :

is unit external normal to any surface dS.
Gauss's Theorem of Divergence :
The surface integral of the normal component of a vector function
taken around a closed surface S is equal to the integral of the divergence of
taken over the volume V enclosed by the surface S.

No comments:
Post a Comment