PH101 Saurabh Basu Class timings (Group II): 9 am-10 am (Wednesdays) 10 am -11 am (Thursdays)

Transcription

1 PH101 Saurabh Basu Class timings (Group II): 9 am-10 am (Wednesdays) 10 am -11 am (Thursdays) Class timings (Group IV): 4 pm- 5 pm (Wednesdays) 3 pm 4 pm (Thursdays) Special: 18 th August and 15 th September (Friday) Class timings (Group II): 11 am-12 noon Class timings (Group IV): 2 pm- 3 pm

2 SYLLABUS upto Mid-Sem.

3 TEXT BOOK

4 Acknowledgement: Some slides and contents are taken from Prof. S.B. Santra & Prof. C.Y. Kadolkar

5

6 VECTORS

7 MATHEMATICAL PRILIMINARIES Definition of vector: A vector is defined by its invariance properties under certain operations -- Translation Rotation Inversion etc

8 A X = A cosθ A y = A sinθ Invariance Under Rotation A X = A cosθ A y = A sinθ A X = Acos(θ φ)=acosθcosφ + Asinθsinφ A y = Asin(θ φ) =A sinθcosφ-acosθsinφ Simplifying A X = A x cosφ + A y sinφ A y = - A x sinφ + A y cosφ

9 In a compact form Transformation equations for the components of a vector can be written as,

10 GENERALISATION TO 3 DIMENSIONS Consider the Rotation Matrix in 3D, R = Rotation about Z-axis by an angle With the help of this we shall prove that റA B is a vector i.e. it is invariant under rotation.

11 Since ԦA is a vector its component transform as, A x = A x cosθ + A y sinθ A y = - A x sinθ + A y cosθ A z = A z (because of rotation about z-axis, the z-component remains invariant.)

12 Similarly B x = B x cosθ + B y sinθ, B y = - B x sinθ + B y cosθ B Z = B Z Now, consider the vector, Consider only x- component (for a moment)

13 (A B) X = (-sinθa x + cosθa y )B Z -(-sinθ B x + cosθb y ) A Z Since A Z = A Z B Z = B Z (A B) X = sinθ(b x A z A z B x ) + cosθ(a y B z - B y A z ) (A B) X = Rx (A B) X Similarly we can prove it for the other components also. (A B) y = Ry (A B) y ; (A B) z = Rz (A B) z Hence, (A B) is invariant under rotation and transforms like a vector.

14 Vector Multiplication Scalar product or Dot product [Remember W = റF. റs] Vector Product or Cross Product [Remember L= റr റp ]

15 Vector Calculus Gradient: To know the direction along which a scalar function changes the fastest φ(x, y, z) is scalar function in cartesian coordinates Gradient operator Find φ for φ (x, y, z) = r =

16 Divergence It quantifies how much a vector function diverges.it is scalar. Example:

17 Curl Circulation of a vector field,

18 Problem 1.11(K &K) Let be an arbitrary vector and be a unit vector in some fixed direction. Show From fig Hence proved.

19 Problem 1.13(K &K) An elevator ascends from the ground with uniform speed. At time a boy drops a marble through the floor. The marble falls with uniform acceleration g = 9.8 m/ and hits the ground sec later. Find the height of the elevator at time At marble reaches ground but

20 Polar Coordinates

21 You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate system.

22 The center of the graph is called the pole. Angles are measured from the positive x axis. Points are represented by a radius and an angl sangle e (r, ) To plot the point 5, 4 First find the angle π/4 Then move out along the terminal side 5

23 Polar Coordinates To define the Polar Coordinates of a plane we need first to fix a point which will be called the Pole (or the origin) and a half-line starting from the pole. This half-line is called the Polar Axis. r θ P(r, θ) Polar Axis A positive angle. Polar Angles The Polar Angle θ of a point P, P pole, is the angle between the Polar Axis and the line connecting the point P to the pole. Positive values of the angle indicate angles measured in the counterclockwise direction from the Polar Axis.

24 More than one coordinate pair can refer to the same point. 210 o 150 o 2 30 o 2,30 o 2,210 o 2, 150 o All of the polar coordinates of this point are: o 2,30 n 360 o o o 2, 150 n 360 n 0, 1, 2...