Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Pie Academy

Physics : Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The document Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
All you need of Physics at this link: Physics

Dot Product

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  are two constant vectors.

Then Dot Product of two vectors is given by 

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 Note: 

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

3. Magnitude of VectorVector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

4. Angle between two vectors Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

5. Also, Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

6. If slopes are given and angle between two curves is θ then tan θ = Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

7. If =   Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev= 0 ⇒ Vectors are orthogonal (θ = 90°)

8. If = Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev= |a||b| ⇒ Vectors are parallel (θ = 0 )

 

Cross Product 

If Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev are two vectors then then cross product between two vectors is given by

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev = unit vector normal to both  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Note:

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

3. Angle between two vectors is θ then sin θ =Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

4.  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

then cross product between two vectors

can be calculated asVector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

5. Geometrically cross product gives the area of triangle

6. If Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevare two sides of triangle then area of the triangle is  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev


Triple Product

1. Geometrically Triple Product gives the Volume of Tetrahedron

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

2. If Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevVector are coplanar vectors


Derivative of a Vector:

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev be a position vector where t is a scalar variable.

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector differentiation is nothing but ordinary differentiation but only difference is Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev position vector.

Formulae:

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev


Vector Operator (∇ - Del)

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevis called vector operator


Gradient 

If ϕ (x, y, z) be a given scalar function thenVector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev is called gradient.

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Note:

1. Physically, gradient gives rate of change of ϕ w.r.t x, y, z separately.
2. Geometrically, it gives normal to the level surface.


Example 1: If ϕ = xyz then find the value of Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev


Note:

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

4. Let ϕ(x,y,z) = c be given equation of the level surface then the outward unique normal vector is defined as  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev


Example 2:

Find the value of unit normal vector  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Solution:

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev


Angle between Two Surfaces

Let ϕ1(x,y,z) = C, ϕ2(x,y,z) = C be given equations of two level surfaces and angle between these two surfaces are given as θ then cos θ Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Note:

The angle between two surfaces is nothing but the angle between their normal.

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev then they are said to be orthogonal surfaces

 

Example 3: 

The angle between the two surfaces x2 + y2+ z2 = 9 and z = x2 + y2 − 3 at the point (2, −1, 2) is

Solution:

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

 

Directional Derivatives of a Scalar Function 

The directional derivative of a scalar function ϕ (x, y, z) in the direction of a vector Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

given asVector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevve then it is in the opposite direction.


Example 4:

The Directional derivative of f(x, y, z) = x2yz + 4xz2 at (1, −2, −1)along (2i – j − 2k) is
Solution:

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Directional Derivative = Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

At (1, −2, −1) we have Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev


Divergence of a Vector

If Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev is a vector point function then Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev  is called Divergence of Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Where  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevare the functions of x, y, z 

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Note:

1. Divergence of a vector is scalar.
2. Physically Divergence measures (outflow - inflow)
3. A vector whose divergence is zero then it is said to be divergence free vector (or) solenoid vector i.e. outflow = inflow = constant 
4. Geometrically, Divergence gives the rate at which the fluid entering in a rectangular parallelepiped per unit volume at the point. 


Curl of a Vector

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevis called the curl of a vector where Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev


Note:

1. If Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev= 0 then it is said to be irrational vector otherwise it is said to be rotational vector.

2. Physically Curl gives the angular Velocity Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

3. Divergence of a curl of any vector is always zero.

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevis known as a vector triple product.


Example 5:

The values of a, b, c so that the vector,

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevis irrotational

Solution: 

Given, that vector V is irrotational 

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

⇒ c = −1, a = 4, b = 2

∴ a = 4, b = 2, c = −1

Dynamic Test

Content Category

Related Searches

Semester Notes

,

Free

,

Vector Calculus - Mathematical Methods of Physics

,

ppt

,

shortcuts and tricks

,

Important questions

,

UGC - NET Physics Physics Notes | EduRev

,

study material

,

mock tests for examination

,

Vector Calculus - Mathematical Methods of Physics

,

MCQs

,

Extra Questions

,

Exam

,

Vector Calculus - Mathematical Methods of Physics

,

practice quizzes

,

UGC - NET Physics Physics Notes | EduRev

,

Viva Questions

,

Objective type Questions

,

pdf

,

video lectures

,

UGC - NET Physics Physics Notes | EduRev

,

Sample Paper

,

past year papers

,

Summary

,

Previous Year Questions with Solutions

;