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Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

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Dot Product

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics  are two constant vectors.

Then Dot Product of two vectors is given by 

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

 Note: 

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

3. Magnitude of VectorVector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

4. Angle between two vectors Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

5. Also, Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

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

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

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

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

 

Cross Product 

If Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics are two vectors then then cross product between two vectors is given by

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics = unit vector normal to both  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Note:

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

3. Angle between two vectors is θ then sin θ =Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

4.  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

then cross product between two vectors

can be calculated asVector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

5. Geometrically cross product gives the area of triangle

6. If Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physicsare two sides of triangle then area of the triangle is  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics


Triple Product

1. Geometrically Triple Product gives the Volume of Tetrahedron

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

2. If Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - PhysicsVector are coplanar vectors


Derivative of a Vector:

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics be a position vector where t is a scalar variable.

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector differentiation is nothing but ordinary differentiation but only difference is Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics position vector.

Formulae:

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics


Vector Operator (∇ - Del)

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physicsis called vector operator


Gradient 

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

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

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 Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics


Note:

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

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 Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics


Example 2:

Find the value of unit normal vector  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Solution:

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics


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 Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Note:

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

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics 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 Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

 

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 Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

given asVector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physicsve 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 Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Directional Derivative = Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

At (1, −2, −1) we have Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics


Divergence of a Vector

If Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics is a vector point function then Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics  is called Divergence of Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Where  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physicsare the functions of x, y, z 

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

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 Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physicsis called the curl of a vector where Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

  Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics


Note:

1. If Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics= 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 Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

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

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physicsis 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 Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physicsis irrotational

Solution: 

Given, that vector V is irrotational 

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics Notes | Study Physics for IIT JAM, UGC - NET, CSIR NET - Physics

⇒ c = −1, a = 4, b = 2

∴ a = 4, b = 2, c = −1

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