18MAB102T
Advanced Calculus and Complex Analysis
Unit II -Vector Calculus
Dr. P. GODHANDARAMAN & Dr. S. SABARINATHAN
Assistant Professor
Department of Mathematics
Faculty of Engineering and Technology
SRM Institute of Science and Technology, Kattankulathur- 603 203.
,Scalar and Vector Fields:
• A physical quantity expressible as a continuous function and which can assum
or more definite values at each point of a region of space, is called point func
the region and the region concerned is called a field.
• Point functions are classified as scalar point function and vector point fu
according as the nature of the quantity concerned is a scalar or a vector.
• At each point P of the field if the function denoted by f(P) is a scalar, it is kn
scalar point function while if 𝑓 𝑃 is a vector, then the function 𝑓 𝑃 is c
vector point function. The concerned field is called a scalar field or a vecto
respectively.
,Example of Scalar Fields:
• The temperature distribution in a medium, the gravitational potential of a syste
of masses and the electrostatic potential of a system of charges.
Example of Vector Fields:
• The velocity of a moving particle, the electrostatic, the magneto static a
gravitational fields.
,Vector Differential Operator DEL 𝜵 :
𝜕 𝜕 𝜕
𝛻= 𝑖 + 𝑗 + 𝑘
𝜕𝑥 𝜕𝑦 𝜕𝑧
Gradient:
Let 𝜙 𝑥, 𝑦, 𝑧 defines a differentiable scalar field. (i.e) 𝜙 is differentiab
each point 𝑥, 𝑦, 𝑧 is a certain region of space. Then the gradient of 𝜙 denote
𝛻𝜙 (or) grad 𝜙 is defined by
𝜕 𝜕 𝜕 𝜕𝜙 𝜕𝜙 𝜕𝜙 𝜕𝜙
𝛻𝜙 = 𝑖 +𝑗 +𝑘 𝜙= 𝑖 +𝑗 +𝑘 = 𝑖
𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑥
,Divergence :
If 𝐹 𝑥, 𝑦, 𝑧 is defined and differentiable vector point function at each point 𝑥
is a certain region of space, then the divergence of 𝐹 denoted by 𝛻. 𝐹 (or) di
defined by
𝜕 𝜕 𝜕 𝜕𝐹
div𝐹 = 𝛻. 𝐹 = 𝑖 +𝑗 +𝑘 𝐹= 𝑖.
𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑥
𝜕 𝜕 𝜕
If 𝐹 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘, then div𝐹 = 𝑖 +𝑗 +𝑘 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘
𝜕𝑥 𝜕𝑦 𝜕𝑧
𝜕𝐹1 𝜕𝐹2 𝜕𝐹3
div𝐹 = + +
𝜕𝑥 𝜕𝑦 𝜕𝑧
,Solenoidal :
If 𝐹 is a vector such that 𝛻. 𝐹 = 0 for all points is a given region, then it is said
a solenoidal vector in that region.
Curl :
If 𝐹 𝑥, 𝑦, 𝑧 is a differentiable vector point function in a certain region of space
the curl or rotation of 𝐹 denoted by 𝛻 × 𝐹 (or) curl 𝐹 (or) rot 𝐹 is defined by
𝑖 𝑗 𝑘
𝜕 𝜕 𝜕
𝛻 × 𝐹 = 𝑐𝑢𝑟𝑙 𝐹 =
𝜕𝑥 𝜕𝑦 𝜕𝑧
𝐹1 𝐹2 𝐹3
, Irrotational :
If 𝐹 is vector such that 𝛻 × 𝐹 = 0 for all points in the region, then it is call
irrotational vector (or) Lamellar vector in that region.
𝛻𝜙.𝑎
Directional derivation :
𝑎
𝛻𝜙
Unit normal vector : 𝑛 =
𝛻𝜙
Angle between the surfaces :
𝛻𝜙1 . 𝛻𝜙2
cos 𝜃 =
𝛻𝜙1 𝛻𝜙2
Advanced Calculus and Complex Analysis
Unit II -Vector Calculus
Dr. P. GODHANDARAMAN & Dr. S. SABARINATHAN
Assistant Professor
Department of Mathematics
Faculty of Engineering and Technology
SRM Institute of Science and Technology, Kattankulathur- 603 203.
,Scalar and Vector Fields:
• A physical quantity expressible as a continuous function and which can assum
or more definite values at each point of a region of space, is called point func
the region and the region concerned is called a field.
• Point functions are classified as scalar point function and vector point fu
according as the nature of the quantity concerned is a scalar or a vector.
• At each point P of the field if the function denoted by f(P) is a scalar, it is kn
scalar point function while if 𝑓 𝑃 is a vector, then the function 𝑓 𝑃 is c
vector point function. The concerned field is called a scalar field or a vecto
respectively.
,Example of Scalar Fields:
• The temperature distribution in a medium, the gravitational potential of a syste
of masses and the electrostatic potential of a system of charges.
Example of Vector Fields:
• The velocity of a moving particle, the electrostatic, the magneto static a
gravitational fields.
,Vector Differential Operator DEL 𝜵 :
𝜕 𝜕 𝜕
𝛻= 𝑖 + 𝑗 + 𝑘
𝜕𝑥 𝜕𝑦 𝜕𝑧
Gradient:
Let 𝜙 𝑥, 𝑦, 𝑧 defines a differentiable scalar field. (i.e) 𝜙 is differentiab
each point 𝑥, 𝑦, 𝑧 is a certain region of space. Then the gradient of 𝜙 denote
𝛻𝜙 (or) grad 𝜙 is defined by
𝜕 𝜕 𝜕 𝜕𝜙 𝜕𝜙 𝜕𝜙 𝜕𝜙
𝛻𝜙 = 𝑖 +𝑗 +𝑘 𝜙= 𝑖 +𝑗 +𝑘 = 𝑖
𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑥
,Divergence :
If 𝐹 𝑥, 𝑦, 𝑧 is defined and differentiable vector point function at each point 𝑥
is a certain region of space, then the divergence of 𝐹 denoted by 𝛻. 𝐹 (or) di
defined by
𝜕 𝜕 𝜕 𝜕𝐹
div𝐹 = 𝛻. 𝐹 = 𝑖 +𝑗 +𝑘 𝐹= 𝑖.
𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑥
𝜕 𝜕 𝜕
If 𝐹 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘, then div𝐹 = 𝑖 +𝑗 +𝑘 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘
𝜕𝑥 𝜕𝑦 𝜕𝑧
𝜕𝐹1 𝜕𝐹2 𝜕𝐹3
div𝐹 = + +
𝜕𝑥 𝜕𝑦 𝜕𝑧
,Solenoidal :
If 𝐹 is a vector such that 𝛻. 𝐹 = 0 for all points is a given region, then it is said
a solenoidal vector in that region.
Curl :
If 𝐹 𝑥, 𝑦, 𝑧 is a differentiable vector point function in a certain region of space
the curl or rotation of 𝐹 denoted by 𝛻 × 𝐹 (or) curl 𝐹 (or) rot 𝐹 is defined by
𝑖 𝑗 𝑘
𝜕 𝜕 𝜕
𝛻 × 𝐹 = 𝑐𝑢𝑟𝑙 𝐹 =
𝜕𝑥 𝜕𝑦 𝜕𝑧
𝐹1 𝐹2 𝐹3
, Irrotational :
If 𝐹 is vector such that 𝛻 × 𝐹 = 0 for all points in the region, then it is call
irrotational vector (or) Lamellar vector in that region.
𝛻𝜙.𝑎
Directional derivation :
𝑎
𝛻𝜙
Unit normal vector : 𝑛 =
𝛻𝜙
Angle between the surfaces :
𝛻𝜙1 . 𝛻𝜙2
cos 𝜃 =
𝛻𝜙1 𝛻𝜙2
