Vector Calculus - Mathematical Methods of Physics, UGC - NET Physics | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

Dot Product

  are two constant vectors.

Then Dot Product of two vectors is given by 

 Note: 

3. Magnitude of Vector

4. Angle between two vectors

5. Also,

6. If slopes are given and angle between two curves is θ then tan θ = 

7. If =   = 0 ⇒ Vectors are orthogonal (θ = 90°)

8. If = = |a||b| ⇒ Vectors are parallel (θ = 0^∘ )

Cross Product 

If  are two vectors then then cross product between two vectors is given by

 = unit vector normal to both  

Note:

3. Angle between two vectors is θ then sin θ =

4. 

then cross product between two vectors

can be calculated as

5. Geometrically cross product gives the area of triangle

6. If are two sides of triangle then area of the triangle is 

Triple Product

1. Geometrically Triple Product gives the Volume of Tetrahedron

2. If Vector are coplanar vectors

Derivative of a Vector:

 be a position vector where t is a scalar variable.

Vector differentiation is nothing but ordinary differentiation but only difference is  position vector.

Formulae:

Vector Operator (∇ - Del)

is called vector operator

Gradient 

If ϕ (x, y, z) be a given scalar function then is called gradient.

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

Note:

4. Let ϕ(x,y,z) = c be given equation of the level surface then the outward unique normal vector is defined as 

Example 2:

Find the value of unit normal vector  

Solution:

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

Note:

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

 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:

Directional Derivatives of a Scalar Function 

The directional derivative of a scalar function ϕ (x, y, z) in the direction of a vector 

given as

ve 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:

Directional Derivative = 

At (1, −2, −1) we have 

Divergence of a Vector

If  is a vector point function then   is called Divergence of 

Where  are the functions of x, y, z 

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

is called the curl of a vector where

 

Note:

1. If = 0 then it is said to be irrational vector otherwise it is said to be rotational vector.

2. Physically Curl gives the angular Velocity 

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

is known as a vector triple product.

Example 5:

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

is irrotational

Solution: 

Given, that vector V is irrotational 

⇒ c = −1, a = 4, b = 2

∴ a = 4, b = 2, c = −1