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 :
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
where 
is a unit vector along tangent of curve PQ
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.
Important Types Of Matrices :
1) Null/ Zero matrix : all elements are zero
2) Diagonal Matrix : square matrix whose all non-diagonal elements are zero.
3) Scalar matrix : diagonal matrix whose all diagonal elements are
equal to zero.
4) Symmetric matrix : In a Square matrix for all i and j , aij = aji
5) Skew Symmetric Matrix : A square matrix is called skew symmetric matrix if
i)for all i and j , aij = - aji
ii) All diagonal elements are zero.
6) Upper Triangular Matrix : A triangular matrix whose elements below the leading diagonal are zero .
7) Orthogonal Matrix : AAT = I
8) Matrix Aθ : = (conjugate of A)T
9) Unitary Matrix : AθA = I
10) Hermitian Matrix : i) aij = conjugate of aji
ii) A = Aθ
11) Idempotent Matrix : A2 = A
12) Periodic matrix : Ak+1 = A
13) Nilpotent Matrix : Ak = 0,k>0
14) Involuntary Matrix : A2 = I
15) Adjoint of A Matrix : adj. A is the transpose of the matrix of co-factors of any matrix A.
16) Inverse Matrix : A-1 = 1/|A| adj. A
** Rank of Matrix :
The rank of matrix is 'r' if
a) It has at least one non-zero minor of order 'r'.
b) every minor of 'A' of order higher than 'r' is zero.
i.e. Rank = No. of non-zero rows in upper triangular matrix.
Rank of augmented matrix A^ defining any given set of linear equations and coefficient matrix C decides type of solution:
1) Consistent Equations : Rank A^ = Rank C
i) Unique solution : Rank A^ = Rank C = n, n => No. of unknowns
ii) Infinite Solution : Rank A^ Rank C = m, m < n
2) Inconsistent Equations : Rank A^ != Rank C No Solution Possible.
Rank of Matrix and Matrix Dependency :
Vectors A1, A2, ..An are said to be dependent if :
1) all vectors are of same order
2) n scalars a1,a2, ..an (at least one non-zero) exist as
a1.A1 + a2.A2 + ...+an.An = 0
Otherwise they are linearly independent.
**Rank of matrix A is maximum no. of linearly independent row vectors of A or no. of linearly independent column vectors of A
Hence A and AT has same rank..
** m vectors with n components each are linearly independent if if the matrix with these vectors as row vectors has rank m, but they are linearly dependent if that rank is less than m.
**TERMS YOU SHOULD ALSO KNOW : **
LINEAR SYSTEM, HOMOGENEOUS AND NON-HOMOGENEOUS SYSTEMS, STOCHASTIC MATRIX, TRIVIAL SOLUTION, GAUSS ELIMINATION, ROW EQUIVALENT SYSTEMS, ECHELON FORM, VECTOR SPACE, ROW SPACE, COLUMN SPACE, CRAMER'S RULE, MINORS AND CO-FACTORS, GAUSS-JORDAN ELIMINATION.
