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_sfj_iTFH14-Bo_EJRZ5NHi_OyX-IuwHrTpieh2P3-ekGQ9HwqcDfWUzS5kiLWTluAITJ7JAMFXOGYHDYbxGJy8ZFACmHO8S5ACzdg4m1N2KvNbHXBjNhLYPDPh3REdL8RG37pM-GDXiVP2OjrBJtwLP3QOb97M9h_vPnVoP8Al0Ik5V8NnwIIxhHIXmiu5a1ZTqSnQ0NjHKhACb5LncK4OsxwKuBQG__e1BzA76lTqFMwmwyeS6yqOfrmXw0-G3QuP5hqcV0F5Oiyu74fAcOmKVySek7giWgexMxlVKK4s1dQ8EhAlK9xbXaFkYbsxUd1auEGUIrr0cRAG1Xz2TlUNNiPFNCkj-qYZ1OqGxL4jQ_kKRO__gvAvQj_wYjVR=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_ugHrnXUa9bU4GbtOcJ1MXptCDqITVI_NsXgQ4on8wZ8Nepm6f-pgmY1ZZYlLhKXd-2YANTvyNjIn3ewfUh4WAy9lJ8FnQvZX7vZ8BkTd2Ol8nC-dyqgP0h7r0u8j9TeY2-vHZiAoW6TLgPFsNVqbIQPQpn2zdHJ-eju40Qn4oBxdKaGz3oSDlAD4gtNawTeZ35k6wFA9YA6ajks_oeh76acjlCJiIvYd94eUzRg2zFoqle64goV9mcG90RombgBw=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_uTXYr9xHj9Hvc6gpwMpd5Ras49fpd_8mXkvV_Qh5KmE_x_dWd98r-YXAL2aHEEWn5Pp4guGQ6URH8FqnVf9k6yxxzZMgRfbJCQkLN1O4RSf9Gchq2q91jaxfih01df3HwNwILUxSfau0mXxkXfjjCwwZgkFgvzsshRRgr28TU1dVX0nHmwKUZeffLoXoRcy27E0B9XNkmfm22wJ9NmbHuwm3DIlauv07H-k89ugELbkHVQmYUQCsQnty-Rv9Gks3ZsPsyZgPX2Vr6GZ9XfkS7Bh-bC098-8RDW6zt13Rq_Mn9rgAjS0L9vH5MsoPOGS1cWCCUbFFjgDcC5P-LP0wxI64GKDPip_8sC8raQTvzZWpCKyAfHoa9HqH_AyxA9KD_LepzL7lA8-gkccBY4-1I=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.

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