4. Vector Differentiation
Let 𝑟 𝑡 = 𝑓(𝑡) then,
𝑓 𝑡+∆𝑡 − 𝑓(𝑡)
If t is a time variable then
represents a velocity vector.
is a vector in direction of tangent to the curve at that point.
2. If 𝑓(𝑡) is constant in magnitude then 𝐹.
3. If 𝑓(𝑡) has constant direction then, 𝐹 ×
5. Vector Differentiation
Vector differential operator :- 𝛻 (nebla)
𝛻 = 𝑖
Gradient of a scalar function :- Let 𝜑(𝑥, 𝑦, 𝑧) be a differentiable scalar point function then
gradient of scalar is denoted by grad 𝜑 or 𝛻𝜑 = 𝑖
Where, 𝛻𝜑 is vector normal to surface 𝜑.
Unit vector normal to surface 𝜑 can be given as
6. Vector Differentiation
Directional derivative :- The directional derivative of differentiable scalar function 𝜑(𝑥, 𝑦, 𝑧)
in the direction of 𝑎 is given by, 𝛻𝜑.
Let 𝑎 = 𝑖, then,
D.D. = 𝛻𝜑.
= ( 𝑖
Angle between surfaces :- It is the angle between the normal to the surfaces at the point of
intersection. Let 𝜃 be the angle between the surfaces 𝜑1 𝑥, 𝑦, 𝑧 = 𝐶1 & 𝜑2 𝑥, 𝑦, 𝑧 = 𝐶2
cos 𝜃 =
7. Vector Differentiation
Divergence of a vector function :- Let 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be a differential vector
point function then,
𝑑𝑖𝑣 𝐹 = 𝛻. 𝐹 =
Note :- If 𝛻. 𝐹 = 0 then 𝐹 is called solenoidal vector.
Curl of a vector function :- 𝑐𝑢𝑟𝑙 𝐹 = 𝛻 × 𝐹 =
𝑖 𝑗 𝑘
𝐹1 𝐹2 𝐹3
Note :- If 𝛻 × 𝐹 = 0 then 𝐹 is called irrotational vector.
If 𝑣 = velocity vector and 𝑤 = angular velocity, 𝑤 =
8. Vector Differentiation
Scalar Potential Function :- If for every rotational vector, a scalar function 𝜑 exist such that
𝐹 = 𝛻𝜑, then 𝜑 is said to be scalar potential function.
1) 𝑐𝑢𝑟𝑙 𝑔𝑟𝑎𝑑 𝜑 = 0
2) 𝑑𝑖𝑣 𝑐𝑢𝑟𝑙 𝐹 = 0
3) 𝑑𝑖𝑣 𝑔𝑟𝑎𝑑 𝜑 = 𝛻 𝛻𝜑 = 𝛻2 𝜑, where 𝛻2=
𝜕𝑧2 (𝛻2 Laplacian Operator)
9. Vector Integration
Line integral :- An integral evaluated over a curve is called line integral.
Let, 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘, be a differentiable point function defined at each point on
curve ‘c’ then its line integral is
𝐹. 𝑑𝑟 =
𝐹1 𝑑𝑥 + 𝐹2 𝑑𝑦 + 𝐹3 𝑑𝑧
If ‘c’ is closed curve 𝑐
Note :- If 𝐹 is irrotational then, the line integral of 𝐹 is independent of path.
When, 𝐹 is irrotational 𝑎
𝐹. 𝑑𝑟 = 𝜑 𝑏 − 𝜑 𝑎 (Where, 𝜑 is scalar potential function)
10. Vector Integration
Green’s theorem :- Let, M(x, y) & N(x, y) be continuous function having continuous first
order partial derivative defined in the closed region R bounded by closed curve ‘c’ then,
(𝑀𝑑𝑥 + 𝑁𝑑𝑦) =
Surface Integral :- Let 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be a differentiable vector point function
defined over the surface S then, its surface integration is
𝐹. d 𝑠 =
𝐹. 𝑛 ds
Where, 𝑛 unit outward drawn normal to the surface
11. Vector Integration
Methods of evaluation of surface integral:-
1. If 𝑅1 is the projection of ‘S’ on to x-y plane then, 𝑠
𝐹 . 𝑛 𝑑𝑠 = 𝑅1
𝐹 . 𝑛
2. If 𝑅2 is the projection of ‘S’ on to y-z plane then, 𝑠
𝐹 . 𝑛 𝑑𝑠 = 𝑅2
𝐹 . 𝑛
3. If 𝑅3 is the projection of ‘S’ on to x-z plane then, 𝑠
𝐹 . 𝑛 𝑑𝑠 = 𝑅3
𝐹 . 𝑛
12. Vector Integration
Let, 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be the differential vector point function defined in volume V,
then its volume integral is 𝑉
Gauss Divergence Theorem :- Let s be a closed surface enclosing a volume V & 𝐹 𝑥, 𝑦, 𝑧 =
𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be the differentiable vector point function defined over S, then,
𝐹. d 𝑠 =
𝑑𝑖𝑣 𝐹 𝑑𝑉
13. Vector Integration
Stoke’s Theorem :- Let S be an open surface bounded by a closed curve ‘c’ & 𝐹 𝑥, 𝑦, 𝑧 =
𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be a differentiable vector function defined over ‘s’, then 𝑐
𝐹 . 𝑑 𝑟 = 𝑠
𝐹. 𝑑 𝑠 = 𝑠
𝛻 × 𝐹 . 𝑛 𝑑𝑠
𝛻 × 𝐹 =
𝑖 𝑗 𝑘
𝐹1 𝐹2 𝐹3