MTH201 Handouts

Table of Contents

Lecture No. 1 Introduction ……………………….………………………………...1 Lecture No. 2 Values of function ……………………… …………………………….5 Lecture No. 3 Elements of three dimensional geometry ………….………………….. 9 Lecture No. 4 Polar co-ordinates ………….…………………………………………..15 Lecture No. 5 Limits of multivariable function……...……………………………….22 Lecture No. 6 Geometry of continuous functions ……..…..…………………………..29 Lecture No. 7 Geometric meaning of partial derivative ….…………………………... 36 Lecture No. 8 Euler theorm……………………...……………………………………. 40 Lecture No .9 Examples ………………………..……………………………………..46 Lecture No. 10 Introduction to vectors ……….. .………………..……………………..51 Lecture No. 11 The triple scalar product or Box product ………………….…………..60 Lecture No. 12 Tangent Planes to the surfaces ………………………………………67 Lecture No. 13 Normal Lines ….……...…………………………………………… 74 Lecture No. 14 Extrema of function of two variables……… ……………………..…..81 Lecture No. 15 Examples …….…………………………… …..……………………..85 Lecture No. 16 Extreme Valued Theorm ……..………… ………………..…………..91 Lecture No. 17 Example………………………………… ……………………………96 Lecture No. 18 Revision Of Integration …………… … …………………………..103 Lecture No. 19 Use of integrals……………………...………………………………..107 Lecture No. 20 Double integral for non-rectangular region ……………………..111 Lecture No. 21 Examples …………………...………………………………………..115 Lecture No. 22 Examples ……………..……………………………………………..119 Lecture No. 23 Polar co-ordinate systems ….………………………………………..122 Lecture No. 24 Sketching ………..…………………………………………………..126 Lecture No. 25 Double Integrals in Polar co-ordinates ….…………….………...…...131 Lecture No. 26 Examples …………..………………………………………….……..135 Lecture No. 27 Vector Valued Functions ………………………………………..…138 Lecture No. 28 Limits Of Vector Valued Functions ……………………………..142 Lecture No. 29 Change Of Parameter ………………….………… ……………….148 Lecture No. 30 Exact Differential …………………………………………………..153 Lecture No. 31 Line Integral ………………………………………...……………...159 Lecture No. 32 Examples ………………………..…………………………………..164 Lecture No. 33 Examples …..………………………………………………………..168 Lecture No. 34 Examples ….………………………………………………………..172 Lecture No. 35 Definite Integrals ……………………………..……………………..179 Lecture No. 36 Scalar Field……………………………………..……………………182 Lecture No. 37 Higher Order Derivative & Leibniz Theorem...…….………..……...186 Lecture No. 38 Taylor and Maclaurin Series …………...……….………..……...196 Lecture No. 39 Numerical Integration ...………...……….………..……...205

INTRODUCTION

Calculus is the mathematical tool used to analyze changes in physical quantities. Calculus is also Mathematics of Motion and Change. Where there is motion or growth, where variable forces are at work producing acceleration, Calculus is right mathematics to apply.

Differential Calculus deals with the Problem of Finding (1) Rate of change (2) Slope of curve

Velocities and acceleration of moving bodies. Firing angles that give cannons their maximum range. The times when planets would be closest together or farthest apart.

Integral Calculus deals with the Problem of determining a function from information about its rates of Change.

Integral Calculus enables us (1) To calculate lengths of curves. (2) To find areas of irregular regions in plane. (3) To find the volumes and masses of arbitrary solids (4) To calculate the future location of a body from its present position and knowledge of the forces acting on it.

Reference Axis System

Before giving the concept of Reference Axis System, we recall you the concept of real line and locate some points on the real line as shown in the figure below, also remember that the real number system consist of both Rational and Irrational numbers that is we can write set of real numbers as union of rational and irrational numbers.

Here in the above figure, we have located some of the rational as well as irrational numbers and also note that there are infinite real numbers between every two real numbers.

Now if you are working in two dimensions, then you know that we take the two mutually perpendicular lines and call the horizontal line as x-axis and vertical line as y-axis and where these lines cut we take that point as origin. Now any point on the x-axis will be denoted by an order pair whose first element which is also known as abscissa is a real number and other element of the order pair which is also known as ordinate will have 0 values, i.e.  , 0 x Similarly any point on the y-axis can be representing by an order pair  0, y . Some points are shown in the figure below. Also note that these lines divide the plane into four regions: First, Second, Third and Fourth quadrants respectively. We take the positive real numbers at the right side of the origin and negative to the left side, in the case of x-axis. Similarly for y-axis and also shown in the figure.

Location of a point

Now we will illustrate how to locate the point in the plane using xand yaxes. Draw two perpendicular lines from the point whose position is to be determined. These lines will intersect at some point on the x-axis and y-axis and we can find out these points. Now the distance of the point of intersection of x-axis and perpendicular line from the origin is the x-coordinate of the point P and similarly the distance from the origin to the point of intersection of y-axis and perpendicular line is the y-coordinate of the point P as shown in the figure below.

In space, we have three mutually perpendicular lines as reference axes, namely x ,y and z axis. Now you can see from the figure below that the planes x = 0 ,y = 0 and z = 0 divide the space into eight octants. Also note that in this case we have (0,0,0) as origin and any point in the space will have three coordinates.

Signs of coordinates in different octants

First of all note that the equation x = 0 represents a plane in the 3d space and in this plane every point has its x-coordinate as 0, also that plane passes through the origin as shown in the figure above. Similarly y = 0 and z = 0 also define a plane in 3d space and have properties similar to that of x = 0 such that these planes also pass through the origin and