KIET GROUP OF INSTITUTIONS, DELHI NCR, GHAZIABAD
B. Tech. (I Sem.), 2020-21
MATHEMATICS-I (KAS-103T)
MODULE 5 (Vector Calculus)
SYLLABUS: Vector differentiation: Gradient, Curl and Divergence and their Physical
interpretation, Directional derivatives, Tangent and Normal planes. Vector Integration: Line
integral, Surface integral, Volume integral, Gauss’s Divergence theorem, Green’s theorem,
Stoke’s theorem (without proof) and their applications.
Course Outcome:
Remember the concept of vector and apply for directional derivatives, tangent and
normal planes. Also for solving line, surface and volume integrals.
Application in Engineering:
Vector calculus plays an important role in computational fluid dynamics or
computational electrodynamics simulation, differential geometry and in the study
of partial differential equations also useful for dealing with large-scale behavior such as
atmospheric storms or deep-sea ocean currents. It is used extensively in physics
and engineering, especially in the description of electromagnetic fields, gravitational
fields and fluid flow.
, CONTENTS
Topic Page No.
1. Vector differentiation (3-22)
1.1 Gradient 3
1.1.1 Geometrical Interpretation of Gradient 4
1.1.2 Properties of Gradient 5
1.2 Directional derivative 7
1.3 Divergence of a vector point function 11
1.3.1 Physical Interpretation of Divergence 12
1.4 Curl of a vector point function 13
1.4.1 Physical Interpretation of Curl 13
1.5 Tangent and Normal Planes 19
2. Vector Integration (23-69)
2.1 Line Integral 23
2.1.1 Work done by a force 23
2.1.2 Circulation 26
2.2 Surface integral 28
2.3 Volume integral 34
2.4 Gauss’s Divergence theorem (without proof) 35
2.4.1 Application of Gauss’s Divergence theorem 41
2.5 Green’s theorem (without proof) 45
2.5.1 Application of Green’s theorem 50
2.6 Stoke’s theorem (without proof) 56
2.6.1 Application of Stoke’s theorem 65
3. E-resources 70
2|P a ge
,1. Vector Differentiation
Introduction: Vector calculus or vector analysis, is concerned with differentiation and
integration of vector fields. It is used extensively in physics and engineering, especially in the
description of electromagnetic fields, gravitational fields and fluid flow.
Point Function: A variable quantity whose value at any point in a region of space depends
upon the position of the point, is called a point function.
Scalar Point Function: If to each point P (x, y, z) of a region R in space there corresponds a
unique scalar f(P), then f is called a scalar point function.
Examples.
(i) Temperature distribution in a heated body,
(ii) Density of a body & (iii) Potential due to gravity.
Vector Point Function: If to each point P (x, y, z) of a region R in space there corresponds a
unique vector f(P), then f is called a vector point function.
Examples.
(i) Forest wind, (ii) The velocity of a moving fluid & (iii) Gravitational force.
1.1 Gradient of a scalar point function
The gradient is closely related to the derivative, but it is not itself a derivative.
The value of the gradient at a point is a tangent vector.
The gradient can be interpreted as the “direction and rate of fastest increase”
3|P a ge
, Vector Differential Operator Del (∇): It is defined as:
𝜕 𝜕 𝜕
𝛻 = 𝑖̂ 𝜕𝑥 + 𝑗̂ 𝜕𝑦+ 𝑘̂ 𝜕𝑧
Gradient of a scalar function: Let ∅(𝑥, 𝑦, 𝑧) be a scalar function, then the vector
𝜕∅ 𝜕∅ 𝜕∅
𝑖̂ 𝜕𝑥 + 𝑗̂ 𝜕𝑦+ 𝑘̂ 𝜕𝑧 is called the gradient of a scalar function ∅.
Thus, 𝑔𝑟𝑎𝑑 ∅ = 𝛻∅
1.1.1 Geometrical Interpretation of Gradient: If a surface ∅(𝑥, 𝑦, 𝑧) = 𝑐
passes through a point P. The value of function at each point of the surface is the same as at P.
Then such a surface is called a level surface through P.
Example. If ø(x,y,z) represent potential at the P. Then equipotential surface ø(x,y,z) = c is a level
surface.
Note: Two level surfaces can’t intersect.
Let the level surface passes through P at which the value of function is ø.
Consider another level surface passing through Q, Where the value of function ø + dø.
Let 𝑟⃗ 𝑎𝑛𝑑 𝑟⃗ + 𝛿𝑟⃗ be the position vector of P and Q then ⃗⃗⃗⃗⃗⃗
𝑃𝑄 = 𝛿𝑟⃗
𝜕𝜙 𝜕𝜙 𝜕𝜙
∇𝜙. 𝑑𝑟⃗ = (𝑖̂ 𝜕𝑥 + 𝑗̂ 𝜕𝑦 + 𝑘̂ 𝜕𝑧 ) . (𝑖̂𝑑𝑥 + 𝑗̂𝑑𝑦 + 𝑘̂ 𝑑𝑧)
𝜕𝜙 𝜕𝜙 𝜕𝜙
= 𝜕𝑥
𝑑𝑥 + 𝜕𝑦 𝑑𝑦 + 𝜕𝑧
𝑑𝑧 = 𝑑𝜙-------- (1)
If Q lies on the level surface of P, then 𝑑𝜙 = 0
From equation (1), we get
∇𝜙. 𝑑𝑟⃗ = 0, then ∇𝜙 ⊥ 𝑡𝑜 𝑑𝑟⃗ (𝑡𝑎𝑛𝑔𝑒𝑛𝑡)
4|P a ge
B. Tech. (I Sem.), 2020-21
MATHEMATICS-I (KAS-103T)
MODULE 5 (Vector Calculus)
SYLLABUS: Vector differentiation: Gradient, Curl and Divergence and their Physical
interpretation, Directional derivatives, Tangent and Normal planes. Vector Integration: Line
integral, Surface integral, Volume integral, Gauss’s Divergence theorem, Green’s theorem,
Stoke’s theorem (without proof) and their applications.
Course Outcome:
Remember the concept of vector and apply for directional derivatives, tangent and
normal planes. Also for solving line, surface and volume integrals.
Application in Engineering:
Vector calculus plays an important role in computational fluid dynamics or
computational electrodynamics simulation, differential geometry and in the study
of partial differential equations also useful for dealing with large-scale behavior such as
atmospheric storms or deep-sea ocean currents. It is used extensively in physics
and engineering, especially in the description of electromagnetic fields, gravitational
fields and fluid flow.
, CONTENTS
Topic Page No.
1. Vector differentiation (3-22)
1.1 Gradient 3
1.1.1 Geometrical Interpretation of Gradient 4
1.1.2 Properties of Gradient 5
1.2 Directional derivative 7
1.3 Divergence of a vector point function 11
1.3.1 Physical Interpretation of Divergence 12
1.4 Curl of a vector point function 13
1.4.1 Physical Interpretation of Curl 13
1.5 Tangent and Normal Planes 19
2. Vector Integration (23-69)
2.1 Line Integral 23
2.1.1 Work done by a force 23
2.1.2 Circulation 26
2.2 Surface integral 28
2.3 Volume integral 34
2.4 Gauss’s Divergence theorem (without proof) 35
2.4.1 Application of Gauss’s Divergence theorem 41
2.5 Green’s theorem (without proof) 45
2.5.1 Application of Green’s theorem 50
2.6 Stoke’s theorem (without proof) 56
2.6.1 Application of Stoke’s theorem 65
3. E-resources 70
2|P a ge
,1. Vector Differentiation
Introduction: Vector calculus or vector analysis, is concerned with differentiation and
integration of vector fields. It is used extensively in physics and engineering, especially in the
description of electromagnetic fields, gravitational fields and fluid flow.
Point Function: A variable quantity whose value at any point in a region of space depends
upon the position of the point, is called a point function.
Scalar Point Function: If to each point P (x, y, z) of a region R in space there corresponds a
unique scalar f(P), then f is called a scalar point function.
Examples.
(i) Temperature distribution in a heated body,
(ii) Density of a body & (iii) Potential due to gravity.
Vector Point Function: If to each point P (x, y, z) of a region R in space there corresponds a
unique vector f(P), then f is called a vector point function.
Examples.
(i) Forest wind, (ii) The velocity of a moving fluid & (iii) Gravitational force.
1.1 Gradient of a scalar point function
The gradient is closely related to the derivative, but it is not itself a derivative.
The value of the gradient at a point is a tangent vector.
The gradient can be interpreted as the “direction and rate of fastest increase”
3|P a ge
, Vector Differential Operator Del (∇): It is defined as:
𝜕 𝜕 𝜕
𝛻 = 𝑖̂ 𝜕𝑥 + 𝑗̂ 𝜕𝑦+ 𝑘̂ 𝜕𝑧
Gradient of a scalar function: Let ∅(𝑥, 𝑦, 𝑧) be a scalar function, then the vector
𝜕∅ 𝜕∅ 𝜕∅
𝑖̂ 𝜕𝑥 + 𝑗̂ 𝜕𝑦+ 𝑘̂ 𝜕𝑧 is called the gradient of a scalar function ∅.
Thus, 𝑔𝑟𝑎𝑑 ∅ = 𝛻∅
1.1.1 Geometrical Interpretation of Gradient: If a surface ∅(𝑥, 𝑦, 𝑧) = 𝑐
passes through a point P. The value of function at each point of the surface is the same as at P.
Then such a surface is called a level surface through P.
Example. If ø(x,y,z) represent potential at the P. Then equipotential surface ø(x,y,z) = c is a level
surface.
Note: Two level surfaces can’t intersect.
Let the level surface passes through P at which the value of function is ø.
Consider another level surface passing through Q, Where the value of function ø + dø.
Let 𝑟⃗ 𝑎𝑛𝑑 𝑟⃗ + 𝛿𝑟⃗ be the position vector of P and Q then ⃗⃗⃗⃗⃗⃗
𝑃𝑄 = 𝛿𝑟⃗
𝜕𝜙 𝜕𝜙 𝜕𝜙
∇𝜙. 𝑑𝑟⃗ = (𝑖̂ 𝜕𝑥 + 𝑗̂ 𝜕𝑦 + 𝑘̂ 𝜕𝑧 ) . (𝑖̂𝑑𝑥 + 𝑗̂𝑑𝑦 + 𝑘̂ 𝑑𝑧)
𝜕𝜙 𝜕𝜙 𝜕𝜙
= 𝜕𝑥
𝑑𝑥 + 𝜕𝑦 𝑑𝑦 + 𝜕𝑧
𝑑𝑧 = 𝑑𝜙-------- (1)
If Q lies on the level surface of P, then 𝑑𝜙 = 0
From equation (1), we get
∇𝜙. 𝑑𝑟⃗ = 0, then ∇𝜙 ⊥ 𝑡𝑜 𝑑𝑟⃗ (𝑡𝑎𝑛𝑔𝑒𝑛𝑡)
4|P a ge
