MTH301 Highlighted Handouts

Table of Contents

Values of function ................................................................................. 7 Elements of three dimensional geometry ............................................. 11 Lecture No. 4 Polar co-ordinates................................................................................ 17 Limits of multivariable function .........................................................24 Lecture No. 6 Geometry of continuous functions ........................................................ 31 Lecture No. 7 Geometric meaning of partial deri.vative .............................................. 38 Lecture No. 8 Euler theorm ........................................................................................ 42 Lecture No .9 Examples ............................................................................................ 48 Lecture No. 10 Introduction to vectors ..........................................................................53 Lecture No. 11 The triple scalar product or Box product ..............................................62 Lecture No. 12 Tangent Planes to the surfaces ............................................................. 69 Lecture No. 13 Noarmal Lines ................................................................................... 76 Lecture No. 14 Extrema of function of two variables… .............................................. 83 Lecture No. 15 Examples............................................................................................ 87 Lecture No. 16 Extreme Valued Theorm .................................................................... 93 Lecture No. 17 Example… ........................................................................................ 98 Lecture No. 18 Revision Of Integration .................................................................... 105 Lecture No. 19 Use of integrals…..............................................................................109 Lecture No. 20 Double integral for non-rectangular region ....................................... 113 Lecture No. 21 Examples.......................................................................................... 117 Lecture No. 22 Examples.......................................................................................... 121 Lecture No. 23 Polar co-ordinate systems .................................................................124 Lecture No. 24 Sketching ........................................................................................ 128 Lecture No. 25 Double Integrals in Polar co-ordinates .............................................. 133 Lecture No. 26 Examples.......................................................................................... 137 Lecture No. 27 Vector Valued Functions ................................................................. 140 Limits Of Vector Valued Functions ................................................ 144 Change Of Parameter ...................................................................... 150 Exact Differential ............................................................................ 155 Line Intagral .................................................................................... 161 Examples ......................................................................................... 166 Examples ......................................................................................... 170 Examples......................................................................................... 174 Definite ........................................................................................... 181 Scalar Field… ................................................................................. 184 Examples......................................................................................... 188 Vector Filed .................................................................................... 192 Periodic Functions… ...................................................................... 197 Fourier Series ................................................................................. 201 Examples ........................................................................................ 206 Examples ........................................................................................ 212 Functions with periods other than 2  ............................................. 217 Laplace Transforms ........................................................................ 222 Theorems….................................................................................... 227 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 locate 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 has 0 values. Similarly any point on the y-axis can be representing by an order pair. 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 x and y axis. 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-C ordinate 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 axis 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.