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_tvPBqpPttJ68ZJFyalsiu0yd06911tS9iKe2EBUNXzvSLYvzpXX1lQHpUAP3hOYAdyf5kuoJTkFw5AsBwxP88z2b0bM3gX9pDDCg1adQwpUt1h1Kx3uLWUnqHaJKQop9xBWNB1tltVwmtCldx-wmhZe7latj-n9RZWvhfnMfoLsoELlPyn5MzWSKAKBDyobzK6e8q7E3dRcqv4tnAWsiBRQhc9aVcFRP1M8NyyQIiJ5GZ6ArLUJbYsfGLMtx3IyQyay-Yrpib6_mlY__wY1LKeD7WnV1xgNWpulz8MfuWTZOhBr1ovc_9bDBbxeMyj6MkatorJkgxWNLgbx56vTCBk_tEtHhjU7E6aYY6GQkEQoK_tG6NDOBFsz9IDVBmB=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_u-N-GsXZkUAbNSmzFUPGcE5mJdynSt0NdQ0nG0HeHrMJIySmSmdpOTdkCjuG3AkoOtjdSzoKkEYVvzj8gl8XzU9nLb0omJrEJDr6jW9XiFtXIUsgcPn56E9TQWhfQvUSNpiRjcF59z0Ka6LVGscZBwrhhZLZxss2fiNz1KsitnYNzMVSJNn8iKvS1b_cCNaqXfPZ1DwlbBabFTg88BBxjIwMbMGYhRapMhBARhv0ytSXy8ZAQW4vOg9Gwc6R7kVw=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_v5GG1FTpavtRkN5NK665_ARelSQDnVmt9TxUg6wbMa8nEW3HXiaKrUJsJwmuQLGc54V00uoIWewVyAYt1h08IpcuOy9uoumKBJRYFN0JabyHamYUam8RKBWf4Trm3kEchNZbNZ9bwFhe573rBGM3034qnPg3J46bjYwqQF4OvjrlH_PECvV4IFOHHgDqOeYPJaVqqz6x8faK5qn_5lUhjlBovExS9nF0O2wO-LNgREvmhyG68jE3hlTf6T4f8iif-9qBHYLp4NnIa3YPoJ94dF-2oYwaNe_esDsdA0KgkgLrhQjqWXYZNMSQZZiPtHZ_bwTobjFfEvdRRaeI3uZB4vBzUlpQYKp9Hzl2EIDRUv5kuqKZCOIfo1Mo3QX0QoaeP0Iz4SI6OqA6RXlbnn6Vg=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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