Chapter 4
Vector Calculus
Vector Differentiation
Vector:
A quantity having both direction and magnitude is called a vector. Eg. Force, velocity
etc. A vector is represented by a directed line segment ⃗⃗⃗⃗⃗
𝐴𝐵. Whose direction
represents the direction of vector and whose length represents the magnitude.
Scalar:
A quantity having only magnitude is called a scalar. They are represented by a real
number. Eg. Mass, length and time.
Vector point function:
If a vector 𝐹 (𝑥, 𝑦, 𝑧) is defined corresponding to each point (x, y, z) of a region, then
𝐹 is called a vector point function.
Scalar point function:
If a scalar (x, y, z ) is defined corresponding to each point (x, y, z) of a region, then
 is called a scalar point function.
Velocity and Acceleration:
Let 𝑟 be the position vector of a moving particle p at any time t, then 𝑣 =
⃗
𝑑𝑣
𝑑2𝑟
𝑑𝑟
𝑑𝑡
defines
the velocity vector of a particle p at time t and 𝑎 =
= 2 defines the acceleration
𝑑𝑡
𝑑𝑡
of a particle at time t.
Vector Differentiation:
⃗ differentiable vector function of scalar t and  is a differentiable scalar
If 𝐴 and 𝐵
function of t then
𝑑
⃗ ) = 𝑑𝐴 +
+ 𝐵
(𝐴
𝑑𝑡
𝑑𝑡
𝑑
𝑑𝐴
⃗
𝑑𝐵
𝑑𝑡
𝑑∅
(∅𝐴) = ∅. 𝑑𝑡 + 𝐴 . 𝑑𝑡
𝑑𝑡
⃗
𝑑
⃗
𝑑𝐵
⃗) = 𝐴𝑋
+
(𝐴 𝑋 𝐵
𝑑𝑡
𝑑𝑡
𝑑𝐴
𝑑𝑡
⃗
𝑋𝐵
1
⃗ ) = 𝐴 . 𝑑𝐵 + 𝐵
⃗ . 𝑑𝐴
.𝐵
(𝐴
𝑑𝑡
𝑑𝑡
𝑑𝑡
Page
𝑑
Chapter 4
Vector Calculus
Vector differential operator ( ∇):
𝜕
𝜕
⃗ 𝜕 is a vector differential operator.
∇ = 𝑖 +𝑗 +𝑘
∇∅ = 𝑖
𝜕𝑥
𝜕∅
𝜕𝑥
+𝑗
𝜕𝑦
𝜕∅
𝜕𝑦
⃗
+𝑘
𝜕𝑧
𝜕∅
If
∅
is a scalar then
𝜕𝑧
Gradient:
If (x, y, z) is a scalar point function then the gradient of  is defined by
∇∅ = 𝑖
𝜕∅
𝜕𝑥
+𝑗
𝜕∅
𝜕𝑦
⃗
+𝑘
𝜕∅
𝜕𝑧
i.e., grad = ∇∅
Directional Derivative:
The directional derivative of a scalar point function  in the given direction is the
rate of change of  in that direction.
If 𝑎 is the unit vector in the given direction then the directional derivative of  is
given by . 𝑎̂ .
Unit vector normal to the surface:
Unit vector normal to the surface  is given by
∇∅
|∇∅|
.
Normal derivative: Normal derivative = |∇∅|
Unit tangent vector: Unit tangent vector =
𝑑𝑟⁄
𝑑𝑡
𝑑𝑟
| ⁄𝑑𝑡|
Angle between two surfaces:
If 1 and 2 are the given two surfaces, then the angle between two surfaces is given
by
∇∅ .∇∅
𝜃 = 𝑐𝑜𝑠 −1 (|∇∅ 1||∇∅2 |)
1
2
Equation of the tangent plane:
The equation f the tangent plane to the surface  at a given point ⃗⃗⃗
𝑟0 is given by
⃗
Where 𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘
⃗
and ⃗⃗⃗
𝑟0 = 𝑥0 𝑖 + 𝑦0 𝑗 + 𝑧0 𝑘
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2
(𝑟 − ⃗⃗⃗
𝑟0 ). 𝑔𝑟𝑎𝑑∅ = 0
Chapter 4
Vector Calculus
Cartesian formula:
(𝑥 − 𝑥0 )
𝜕∅
𝜕𝑥
+ (𝑦 − 𝑦0 )
𝜕∅
𝜕𝑦
+ (𝑧 − 𝑧0 )
𝜕∅
𝜕𝑧
=0
Equation of the normal:
The vector equation of the normal at a given point ⃗⃗⃗
𝑟0 on the surface  is
(𝑟 − 𝑟⃗⃗⃗0 )𝑋𝑔𝑟𝑎𝑑∅ = 0
Cartesian formula:
𝑥 − 𝑥0
𝜕∅
𝜕𝑥
=
𝑦 − 𝑦0
𝜕∅
𝜕𝑦
=
𝑧 − 𝑧0
𝜕∅
𝜕𝑧
Divergence:
If 𝐹 (𝑥, 𝑦, 𝑧) is a vector point function continuously differentiable in a given region
of a space then the divergence of 𝐹 is given by
∇. 𝐹 = 𝑖.
𝜕𝐹
𝜕𝑥
+ 𝑗.
⃗
𝜕𝐹
𝜕𝑦
𝜕𝐹
+ ⃗⃗⃗
𝑘.
𝜕𝑧
i.e., div(𝐹 ) = ∇. 𝐹
Solenoidal Vector:
A vector 𝐹 is called solenoidal if div(𝐹 ) = 0.
Curl:
If 𝐹 (𝑥, 𝑦, 𝑧) is a vector point function continuously differentiable in a given region
of a space then the curl of 𝐹 is given by
∇𝑋𝐹 = 𝑖𝑋
𝜕𝐹
𝜕𝑥
+ 𝑗𝑋
𝜕𝐹
𝜕𝑦
⃗ 𝑋 𝜕𝐹
+ 𝑘
𝜕𝑧
Page
Irrotational Vector:
A vector 𝐹 is called irrotational if curl(𝐹 ) = 0.
3
i.e., curl(𝐹 ) = ∇𝑋𝐹
Chapter 4
Vector Calculus
Vector Identities:
div(∅. 𝐹 ) = ∅𝑑𝑖𝑣𝐹 + 𝐹 . 𝑔𝑟𝑎𝑑∅
i.e., ∇. (∅𝐹 ) = ∅(∇. 𝐹 ) + 𝐹 . (∇∅)
curl(∅. 𝐹 ) = ∅𝑐𝑢𝑟𝑙𝐹 + 𝑔𝑟𝑎𝑑∅ 𝑋𝐹
i.e., ∇𝑋(∅𝐹 ) = ∅(∇𝑋𝐹 ) + (∇∅)𝑋𝐹
⃗)= 𝐵
⃗ . (𝑐𝑢𝑟𝑙𝐴) − 𝐴. 𝑐𝑢𝑟𝑙𝐵
⃗
div(𝐴𝑋𝐵
⃗)=𝐵
⃗ . (∇𝑋𝐴) − 𝐴. (∇X𝐵
⃗)
i.e., ∇. (𝐴𝑋𝐵
⃗ ) = 𝐴(𝑑𝑖𝑣𝐵
⃗)− 𝐵
⃗ 𝑑𝑖𝑣𝐴 + (𝐵
⃗ . ∇)𝐴 − (𝐴. ∇)𝐵
⃗
curl(𝐴𝑋𝐵
⃗ ) = 𝐴(∇. 𝐵
⃗)−𝐵
⃗ (∇. 𝐴) + (𝐵
⃗ . ∇)𝐴 − (𝐴. ∇)𝐵
⃗
i.e., ∇𝑋(𝐴𝑋𝐵
curl(grad) =0
div(curl𝐴 )= 0
𝜕2 ∅
𝜕2 ∅
𝜕𝑥
𝜕𝑦
𝜕𝑧 2
+
2
+
2
𝜕2
𝜕2
𝜕2
𝜕𝑥
𝜕𝑦
𝜕𝑧 2
+
2
+
2
is called the Laplacian operator.
4
Note: ∇2 =
𝜕2 ∅
Page
div(grad) =
Chapter 4
Vector Calculus
Vector Integration:
If 𝐹 (𝑡) and 𝑓 (𝑡) are two vector functions of a scalar variable t such that
𝑑
𝐹 (𝑡) = 𝑓 (𝑡) then ∫ 𝑓 (𝑡)𝑑𝑡 = 𝐹 (𝑡) + 𝑐
Where 𝑐 is arbitrary constant vector.
𝑑𝑡
Line integral:
An integral which is evaluated along a curve is called a line integral. If A and B are
two points on the curve C, then the tangential line integral of 𝐹 along the curve is
𝐵
∫𝐴 𝐹 . 𝑑𝑟⃗⃗ = ∫𝐶 𝐹 . 𝑑𝑟
Circulation:
If C is a simple closed curve then the tangential line integral of F around C is called
the circulation of F about C and is written as ∮𝐶 𝐹 . 𝑑𝑟 .
Work done by a force:
If 𝐹 (x, y, z) is a force acting on a particle which moves along the given curve C,
then the total work done by a force 𝐹 is defined by ∫𝐶 𝐹 . 𝑑𝑟 .
Surface Integral:
An integral which is evaluated over a surface is called a surface integral and is denoted
by
∬𝑆 𝑓(𝑥, 𝑦, 𝑧)𝑑𝑆
Where f(x, y, z) is a single valued function of position defined over S.
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𝐹 . 𝑛̂ 𝑑𝑆
∬𝑆 ⃗⃗⃗
5
Flux:
Consider any point p on the surface and let 𝑛̂ be the unit vector at p in the direction
⃗⃗⃗ . 𝑛̂ over the surface S
of the outward normal to the surface. Then the integral of 𝐹
is called the flux of ⃗⃗⃗
𝐹 over S and is denoted by
Chapter 4
Vector Calculus
Volume Integral:
An integral which is evaluated over a volume bounded by a surface is called a volume
integral or sometimes called a space integral and is denoted by
∭𝑉 ∅(𝑥, 𝑦, 𝑧)𝑑𝑉
Where ∅(𝑥, 𝑦, 𝑧) be a scalar field in the region V.
Green’s theorem:
Let R be a closed bounded region in the xy plane whose boundary C consists of
finitely many smooth curves. Let M and N be continuous functions of x and y having
𝜕𝑀
𝜕𝑁
continuous partial derivatives
𝑎𝑛𝑑
in R, then
𝜕𝑦
𝜕𝑁
∬𝑅 ( 𝜕𝑥 −
𝜕𝑀
𝜕𝑦
𝜕𝑥
) 𝑑𝑥𝑑𝑦 = ∮𝐶 (𝑀𝑑𝑥 + 𝑁𝑑𝑦)
Gauss Divergence Theorem:
The surface integral of the normal component of a vector function F over a closed
surface S enclosing volume V is equal to the volume integral of the divergence of F
taken throughout the volume V.
𝐹 . 𝑛̂𝑑𝑆 = ∭𝑉 ∇. ⃗⃗⃗
𝐹 𝑑𝑉
∬𝑆 ⃗⃗⃗
Stoke’s theorem:
The surface integral of the normal component of the curl of a vector function F over
an open surface S is equal to the line integral of the tangential component of F around
the closed curve C bounding S.
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6
⃗⃗⃗ . 𝑑𝑟⃗⃗ = ∬ ∇𝑋𝐹
⃗⃗⃗ . 𝑛̂𝑑𝑆
∫𝐶 𝐹
𝑆