Class notes for Course 18MAB102T - Advanced Calculus and Complex Analysis Our notes deliver a thorough exploration of vector calculus principles, including gradient, divergence, and curl, as well as the crucial theorems such as Green's, Stokes', and the Divergence Theorem. Designed for both clarit...
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
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
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