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Leibnitz Theorem: We know that the product of two n times differentiable functions is n times differentiable as well. The Leibnitz formula gives an explicit expression for the n-th derivative of the product

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Differential Calculus-IA: Case -3 If m < n in (*)

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Problems on Leibnitz Theorem-2: 10. If show that

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Radius of Curvature: Let Pbe any point on the curve C. Draw the tangent at Pto the circle. The circle having the same curvature as the curve at P touching the curve at P, is called the circle of curvature. It is also called the osculating circle. The centre of the circle of the curvature is called the centre of curvature. The radius of the circle of curvature is called the radius of curvature and is denoted by ‘ρ’.

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Taylor’s Theorem: Taylor's Theorem for a function of a single variable and Maclaurin's series functions. Suppose a function f (x) satiesfy the following conditions: 1. f (x), f' (x), f'' (x),........ f ^(n-1) (x) are continuous in the closed intervals [a,b]

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Curve Tracing: The elementary method of tracing (or plotting) a curve whose equation is given in rectangular coordinates, and one with which the student is already familiar, is to solve its equation for y (or y ), assume arbitrary values of y (or y ), calculate the corresponding values of y (or y ), plot the respective points, and draw a smooth curve through them, the result being an approximation to the required curve. This process is laborious at best, and in case the equation of the curve is of a degree higher than the second, the solved form of such an equation may be unsuitable for the purpose of computation, or else it may fail altogether, since it is not always possible to solve the equation for y or y. The general form of a curve is usually all that is desired, and the Calculus furnishes us with powerful methods for determining the shape of a curve with very little computation. The first derivative gives us the slope of the curve at any point; the second derivative determines the intervals within which the curve is concave upward or concave downward, and the points of inflection separate these intervals; the maximum points are the high points and the minimum points are the low points on the curve. As a guide in his work the student may follow: Rule for tracing curves. Rectangular Coordinates:

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Partial Derivatives: Given a function of one variable say f(x); we define the derivative of f(x) at x = a to be

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Problems on Differential Calculus I-1: 9.

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Euler-Lagrange Equation: Theorem 1: If y (x) is a curve in which minimize the functional

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Problems on Radius of Curvature: 1. Find the radius of curvature at any point on the curve y = a log sec ( x/a ). Solutions:

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Cauchy’s Mean Value Theorem: If two function f (x) and g (x) are such that: 1. f (x) and g (x) are continuous in the closed intervals [a,b]

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Radius of Curvature in Parametric Form: Let x = f (t) and y = g (t) be the Parametric equations of a curve C and P ( x, y) be a given point on it. Then

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Special Cases: Theorem (change of variable theorem: Polar Coordinates). Let x = r cosθ, y = r sinθ the inverse function are:

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Problems on Fundamental Theorem-1: 3. Verify Rolle's Theorem for the functions.

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Radius of Curvature in Polar Form: Let r = f (θ) be the equations of the curve in the polar form and p(r, θ) be a point on it. Then we know that Differentiate w.r.t r we get;

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Vector Functions: A Vector - Valued function is a vector function whose components are single - valued functions (Scaler Valued Function). For example, given three single valued function f₁(t) , f₂(t), f₃(t) we can form the vector valued functions:

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Differential Operators: In this section, we shall express the gradient, divergence and curl in terms of orthogonal curvilinear coordinates u₁, u₂, u₃. Then, the Laplacian can be expressed as the divergence of a gradient by the chain rule, we have

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Gauss Divergence Theorem: Statement: If V is the volume bounded by a surface S and is continuously differentiable vector function then.

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Green's Theorem: This theorem gives the relation between the plane, surface and the line integrals. Statement: If R is a closed region in the xy-palne bounded by a simple closed curve C and M (x,y) and N (x,y) are continuous function having the partial derivatives in R then,

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Vector Line Integral: If is the force acting on a particle at a point P whose positive vector is r on a curve C then represent physically the total work done in moving the particle along C. Thus total work done is;

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Stoke Theorem: Statement: If S is a surface bounded by a simple closed curve C and if is any continuously differentiable vector function then,

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Curl of Vector: Let,

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Vector Gradient: Basic properties of Vector Gradient are given as follows: 1. If f and g are two scaler point function that Solution: We have

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Directional Derivative of Vector: The derivative of a point function (scalar or vector) in a particular direction is called its directional derivative along the direction. The directional derivative of a scalar point function Φ in a given direction is the rate of change of Φ in the direction. It is given by the component of grad Φ in that direction. The directional derivatives of a scaler point function Φ (x,y,z) in the direction of . Directional derivatives of Φ is maximum in the direction of Φ. Hence the maximum directional derivatives is .

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Problems on Vector Calculus-2: 7. Using divergence theorem, evaluate and S is the surface of the solid cut off by the plane x y z = a from the first octant. Solution: Now

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Divergence of Vector: We now derive the expression for the divergence of a vector. In this orthogonal curvilinear coordinates system (u₁, u₂, u₃) the unit vector can be expressed as

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Problems on Integrals Theorem-3: 1. Using divergence theorem, evaluate and S is the surface of the solid cut off the plane x y z = a from the first octant. Solution: Now

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Surface and Volume Integrals: Surface Integral: An integral evaluated over a surface is called a Surface Integral. Consider a surface S and a point P on it. Let be a vector function of x,y,z defined and continuous over S. In is the unit outward normal to the surface S and P then the integral of the normal component of at P over the surface S is called the Surface integral written as

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Orthogonal Curvilinear Coordinates: Definition: Let the coordinates (x,y,z) of any points be expressed as function of (u₁,u₂,u₃) so that x = x (u₁,u₂,u₃), y = y (u₁,u₂,u₃), z = z (u₁,u₂,u₃) then u,v,w can be expressed in terms of x,y,z as u₁ = u (x,y,z), u₂ = v (x,y,z), u₃ = w (x,y,z). And also if then the system of coordinates (u₁,u₂,u₃) will be an alternatives specification of the cartesian system (x,y,z) and (u₁,u₂,u₃) are called the Curvilinear Coordinates of the points. If one component is kept constant say u₁ = c, then and the locus of (x,y,z) is a surface is called a coordinates surface. Thus we have three families of coordinate system corresponding to u₁ = c, u₂ = c, u₃ = c. Suppose u₁ = c, u₂ = c, u₃ ≠ c. in that case locus is obtained is called a coordinates curve and also there are such families.

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Theorem of Line Integral: One way to write the Fundamental Theorem of calculus is:

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Extrema of Function of Several Variables: Definition: We call f (a,b) a relative (local) maximum if there is an open Disk D centered at (a,b) for which f (a,b) ≥ f (x,y) for all (x,y) ∈ D. Similarly f (a,b) is called a relative (local) minimum there is an open Disk D for which f (a,b) ≤ f (x,y) for all (x,y) ∈ D. In either case f (a,b) is called a relative (local) extremum of f. Definition: The point (a,b) is called a critical point of f if (a,b) is in the domain of f and either f = 0 or is undefined. Note that f = 0 if f_x (x,y) = 0 and f_y (x,y) = 0. f doesnot exist.

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Indeterminate Forms: If f (x) and g (x) are two functions, then we know that

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Problems on L' Hospital Rule-1: 5. Evaluate

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Polar Curves-2: 1. Derive an expression for arc length in Cartesian form. Solution: Let P (r,θ) and Q (r δr, θ δθ) be two neighbouring point on the graph of the functions r f (θ). So that they are at length S and S δs from a fixed point A on the curve.

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Indeterminate Forms ∞ – ∞ and 0 × ∞: L' Hospital rule can be applied to limits which take the indeterminant forms First we transform the given limit in the form of and then by use L' Hospital rule to evaluate the limits.

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Problems on Taylor's Theorem: 1. Expand ex sin y by using Maclaurin's Theorem upto the third degree terms. Solution: Let

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Problems on Lagrange’s Method of Undetermined Multipliers: 1. Find the minimum value of x² y² z² subject to the conditions ax by cz = p. Solutions: Let

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Lagrange’s Method of Undetermined Multipliers: So far, we have considered the method of finding the extreme values of a function f (x, y, z), where these variables x, y, z are independent. Sometimes we may have to find the maximum or minimum values of a function f (x, y, z), when x, y, z are connected by some relations say φ (x, y, z) = 0. In such cases we may eliminate z from the given conditions and express the function f (x, y, z) as a function of two variables x and y and obtain the extreme values of f as before. In such cases we have an alternative method for finding the critical points called “Lagrange’s method of undetermined multipliers”. Suppose we want to find the maximum and minimum values of the functions

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Problems on Differential Calculus II: 2. Evaluate Apply L's hospital Rule;

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Taylor’s Theorem For Function of Two Variables: Statements: If f (x,y) has continuous partial derivatives up to nth order in a neighbourhood of a point (a,b) then

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Jacobian of Transformation: In this section, we explore the concept of a "Derivative" of a coordinate transformation, which is known as the Jacobian of the transformation. However, in this course, it is the determinant of the Jacobian that will be used most frequently. If we let u = < u,v >, p = < p,q >, and x = < x,y >, then (x,y) = T (u,v) is given in vector notation by X = T (u). This notation allow us to extend the concept of a toatal derivatives to the total derivatives of a coordinate transformation.

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Problems on Polar Curve-2: Solution of 2:

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Problems on Various Indeterminate Forms: 3. Evaluate

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Problems Maxima And Minima of Function of Two Variables-1: 3. Find the extreme value of the functions x³ y³ - 3xy. Solutions: Let

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Maxima And Minima of Function of Two Variables: Maximum Value of Function: A function f (x,y) is said to have a maximum point (a,b), if these exists a neighbourhood N of (a,b) such that;

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Introduction of Matrices: A rectangular array of number is called a matrix. Each number is called the element or entry of the matrix. A Row of a matrix is a horizontal array and a column is a vertical one. The size or order of a matrix is determined by the number of rows and column of that matrix. A matrix is said to be of order or size m n if it has m rows and n column. If m = n, the matrix is called the square matrix of order or size n n. Any real number is called a scalar. There are a number of ways to represent an m n matrix.

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Problems on Matrices: Example-1 Consider the quadratic form:

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Matrix Multiplication: Let A = [ a_ij ]_mn and B = [ b_ij ]_np then the product AB = C = [ c_ij ]_mp where

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Properties of Matrices: Basic properties of Matrices which are given as: 1. Zero matrix or Null matrix: A matrix whose all element equal to zero.

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Eigenvalues of Matrices: The eigenvalue are also called characteristics roots, proper values, latent roots, etc. These have fundamental importance is understanding the properties of a matrix. For arbitrary square matrices, eigenvalues are complex root of the characteristics polynomials. After all not all polynomials have real roots. For example λ2 1 = 0 ahs roots λ1 = i and λ2 = -i where i denotes The eigenvalues are denoted by λi where i = 1,2......,n are in non - increasing order of absolute values. For complex number the absolute value is defined as the square root of the product of the number and its complex conjugate. The complex conjugate of i is -i, their product is 1 with square root also 1. Thus lil = 1 and l -il = 1 and the polynomials holds true.

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Scalar Multiplication: Let A = [ a_ij] _mn and r be a real number. Then the Scalar Multiplication of the matrix A and the scalar r is a matrix C such that C = [ C_ij] _mn where C_ij = r a_ij for all i and j. Example:

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Transpose of Matrix: If A = [ a_ij ] is an m n matrix, then the transpose of A is A^T = [ a_ij ] which is an n m matrix. Thus the transpose of A is obtained from A by interchanging the rows and column of A. We obtain

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Echelon Form of Matrix: An m n matrix A is said to be in reduced row echelon form if it satiesfy the following properties: 1. All zero rows, if there are any appear at the bottom of the matrix.

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Method of Finding Eigenvalues and Eigenvectors: Let A be an n × n matrix: 1. To find the eigen value of A solve the characteristics equations

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Determinant: Let then the determinant of the matrix A which is denoted by det (A) or |A| is defined by det (A) = a₁₁a₂₂ - a₂₁a₁₂ . Example-1:

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Rank and Trace of Matrix: An T×p matrix X is said to be of rank p if the dimension of the largest nonsingular square submatrix is p. (Recall that nonsingular means nonzero determinant, and X is often the matrix of regressors with T observations). The idea of the rank is related to linear independence as follow. We have rank(A)= min[ Row rank(A), Column rank(A)],

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Cayley-Hamilton Theorem: A linear transformation T from avector space V to itself ( T : V → V ) is called Linear operator on V. Theorem: Let T : V → Vbe a linear operator on a finite dimensional vector space V. Let p be the charecteristics polynomial of T. Then p (T) = 0. Proof : Choose a basic β = {v₁,.......,v_n} for V. I will show that p(T) = 0 by showing that p (T)vi = 0 for all i.

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Nonsingular Matrix: A square n n matrix A is said to be non singular or invertible if there is another n n matrix B such that;

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System of Linear Equation: An equation of the form;

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Properties of Determinants: Basic properties of Detaerminants are given as follows: 1. If A is a square matrix then det (A) = det (A^T).

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Idempotent Matrices: In ordinary algebra the familiar number whose square is itself are: 0 and 1. Any integer power of 1 is 1. In matrix algebra, the identity matrix I plays the role of the number 1 and I certainly have the property Iⁿ = I. There are other non null matrices which are not identity and yet have this property namely A2 = A. Hence a new name "idempotent" is needed to describe them. For example the hat matrix is defined as means that matrix is idempotent . Verify that for any integer power n. If eigenvalues are the relation means that Since the only number whose nth power is itself are unity and zero. it is plausible that the eigenvalue of an idempotent matrix are 0 and1. Recall that the rank of matrix equals the number of nonzero eigenvalue and that the trace of matrix is the sum of eigenvalue. Using these two results it is clear that for idempotent matrices that G whose eigenvalue are 0 and 1 we have the interresting relation trace (G) = rank (G).

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Unitary Matrices: A n n matrix A and B are unitarily similar if there exists a unitary matrix U such that U*AU = B. Theorem: (Schur's Theorem) Every square matrix is unitary similar to an upper traingular matrix (in Cⁿ).

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Solution to Linear System by Inverse Method: If A is an n n matrix then the linear system AX = B is a system of n equations in m unknown. Suppose that A is a non-singular. Then A^-1 exists and we can multiply, AX = B

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Diagonalisation of Matrices: Two matrices A,B ∈ M_n(K) are congruent if there is an invertible matrix P ∈ GL_n(K) such that,

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Solution to a Linear System: Gaussian Elimaination and Gauss-Jordan Reduction: Given the system AX = B to solve this system by Gaussian alimination method we transform the argumented matrix [A|B] to the matrix [C|D] which is row echelon matrix by using elementry row operations. Then to find the solutions of the system from the corresponding argumented matrix [C|D] we back substitute.

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Multiple Integrals: In this topic we discuss a repeated process of integration of a function of two and three variables reffered to as:

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Problems on Beta and Gamma Functions: 1. Evaluate the following:

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Dirichlet Integral and Fourier Series: Many of the ideas used in the previous section arise naturally in the basic analysis of Fourier Series. We introduce the basuc definition and then prove a theorem that implicity contains the solution to the Dirchlet integrals. If f is any function that is integrable on [ -π, π], the numbers

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Problems on Multiple Integral: 1. Evaluate

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Beta And Gamma Function: Definitions: 1. The Beta function denoted by B (m,n) or β (m,n) is defined by

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Dirichlet Integral: The integral we work with here is

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Applications to Area and Volume: 1. Area of the region R in the Cartesian form. 2. Area of the region R in the Polar form.

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Triple Integrals: Properties: 1. Volume Property:

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Double Integral by Changing the Order of Integration: In the evaluation of the double integrals sometimes we may have to change the order of integration so that evaluation is more convenient. If the limits of integration are variables then change in the order of integration changes the limits of integration. In such cases a rough idea of the region of integration is necessary. Double Integral by Change of Variables:

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Objective Question: 1. The area bounded by the curves y² = x - 1 and y = x - 3

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Problems on Integrals-1: 7. Find the area of the circle x² y² = a². by using the double integrals. Solution: Since the circle is symmetric about the coordinates axes, area of circle is 4 times the area OAB as shown in the figure. For the region OAB, y varies from 0 to and x varies from 0 to a.

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Problems on Applications to Area and Volume-3: Type-2: Evaluation of Double Integral by changing the order of integration: 1. Change the order of integration and hence evaluate

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Relationship between Beta and Gamma Functions: The Beta and Gamma function are related by

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Triple Integrals using Cylindrical Coordinates: Let T be a solid whose projection onto the xy-plane is labelled Ωxy. Then the solid T is the set of all points (x,y,z) satiesfying

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Problems on Dirichlet Integrals: Example1. First we note that with λ = 1 we recover the arc tangent formula