Differential Calculus-II

Jacobian matrix of partial derivatives Let 'u' and 'v' be functions of two independant variables 'x' and 'y'. The jacobian(J) of 'u' and 'v' with respect to 'x' and 'y' is symbolically represented and defined as follows Jacobian matrix problems If u=yz/x, r=zx/y, w=xy/z, Show that J [(u, v, w)/ (x, y, z)] =4 If u=x+3y^2, v=4x^2 yz, w=2z^2-xy, Evaluate del (u,v,w)/del (x,y,z) at the point (1,-1, 0) If u=(x_2 x_3)/x_1, v=(x_1 x_3)/x_2, w=(x_1 x_2)/x_3 find the value of Jacobian…

Maxima and Minima for a function of two variables 1. A function f(x) is said to have a maximum value at a point x=a if there exists a neibourhood of the point 'a' say (a+h), h is small, such that f(a)>f(a+h). 2. Similarly if f(a)<f(a+h) then f(x)is said to have minimum value at x=a. 3. A necessary condition for f(a) to be an extreme value of f(x) is that f'(a)=0 f(a)is maximum if f'(a)=0 and f''(a)<0 f(a)is minimum if f'(a)=0…

LaGrange's method of multipliers with one subsidiary condition Procedure Step 1: Auxilliary function is formed by Step 2: Form equations Fx=0 ,Fy=0 and Fz=0 where Fx is the partial derivative of ‘F’ with respect to ‘x’ Fy is the partial derivative of ‘F’ with respect to ‘y’ and Fz is the partial derivative of ‘F’ with respect to ‘z’. Step 3: Solve for (x,y,z) and 'λ',the values of u(x,y,z)are the Stationary values. LaGrange's Method Problems Find the minimum value of…

Differentiation of composite functions If u=f(x_1 y)then total differential equation of 'u' is given by Type 1 : Total derivative rule If u=f(x,y) where x=x(t) and y=y(t),then Type 2 : Total derivative Chain rule If u=f(x,y) where x=x(r,s) and y=y(r,s),then u is a function of (x,y),x and y are dependant on (r,s),hence u becomes a function depndant on (r,s) Total Derivatives Problems Find du/dt when u=x^3 y^2+x^2 y^3 with x=at^2, y=2at. Use partial derivatives. Find du/dt if u=xy+yz+zx and x=t…

Homogenous function A function u=f(x,y) is said to be homogenous function of degree ‘n’ if it can be expressed in the form x^n g(y/x) or y^n g(x/y), ‘g’ being any arbitrary function. Similarly a function u=f(x,y,z)is said to be a homogenous function of degree ‘n’ if it can be expressed in the form Euler’s theorem on homogenous function Statement: If  is a homogenous function of degree ‘n’ then Proof :   Euler’s theorem Problems If u=(x^3+y^3)/sqrt(x+y) prove that x du/dx…

Standard form of a linear equation and its solution A differential equation is said to be linear if the dependent variable and its derivative occurs in the first degree only and they are not multiplied together. A differential equation of the form Bernoulli’s differential equation The Differential equation of the form Where ‘P’ and ‘Q’ are functions of ‘x’ is called Bernoulli’s differential equation in ‘y’, then divide the entire equation by  and then by using variable substitution method convert…

Exact Differential Equations An exact differential equation is a type of ordinary differential equation (ODE) in which the total differential of a function is equal to the product of a function and a differential. Exact differential equations have the property that their solutions are unique, provided suitable initial or boundary conditions are specified. TYPE 1 For any given function f(x,y), the necessary and the sufficient condition for the differential equations M (x,y)dx+N(x,y)dy=0 to be an exact equation is Type 2…

Partial derivatives deals with the differentiation of a function of many independent variables. Partial derivatives Examples: Area of rectangle depends on its length and breadth which is defined as A(l,b) Volume of parallelepiped depends on its length, breadth and height which is defined as V(l,b,h) Where equation (1) -> partial derivative of ‘u’ with respect to ‘x’, ‘y’ being a constant variable and (2) -> partial derivative of ‘u’ with respect to ‘y’, ‘x’ being a constant variable. Type 1:…

In-determinate Forms If f(x) at x = a assumes forms like etc., which do not represent any value are called In-determinate forms. The concept of limit gives a meaningful value for the function  at overcomes these in-determinate forms. The differentiations for such forms are performed using L'Hospital’s (French Mathematician) rule. L’ Hospital’s theorem Statement If  f(x) and g(x) are two functions such that Problems are categorized under three different types. Type 1 Type 2 Type 3 Note Standard limits that…

Maclaurin’s Theorem, Maclaurin’s Series, Maclaurin’s Series Problems and Solutions Maclaurin’s Theorem, Maclaurin’s Series, Maclaurin’s Series Problems and Solutions Problems Using Maclaurin's series, prove that √(1+sin⁡2x )=1+x^2/2-x^3/6+x^4/24+….. Find the Maclaurin's expansion of log(sec x) up to x^4 terms Obtain the Maclaurin’s expansion of the function log(1+x) up to 4th degree terms Expand log(1+cos x ) by Maclaurin's series up to the term containing x^4 Expand e^x/(1 + e)^x using Maclaurin’s series up to and including 3rd degree terms Expand the Maclaurin's…

Taylor’s Theorem, Taylor’s Series, Taylor’s Series Problems and Solutions Taylor’s Theorem, Taylor’s Series, Taylor’s Series Problems and Solutions Problems Find the Taylor’s series of sin⁡x in powers of (x- π/2) up to the fourth degree term Find the Taylor’s Series of log x in powers of (x -1) up to fourth degree terms Expand tan x in Taylor’s Series up to three in powers of (x-π/4) Expand log(cos x) about the point x=π/3 up to 3rd degree terms using Taylor’s…