Faculty of Engineering and Technology.
Department of Applied Science & Humanities.
Subject: Mathematics –II (303191151)
Semester: 2nd Sem. B.Tech Programme (All Branches)
Lecture Notes: Unit – 5, Vector Calculus
Scalar point function:
• If corresponding to each point 𝑃(𝑥, 𝑦, 𝑧) of a region 𝑅 in space there
corresponds a unique scalar function 𝑓 = 𝑓 (𝑥, 𝑦, 𝑧) then, 𝑓 is called a scalar
point function and R is called a scalar field.
• For example,
(i) the temperature field in a body.
(ii) The pressure field of the air in the earth’s atmosphere.
(iii)The density of a body.
These quantities take different values at different points.
Note: A scalar field which is independent of time is called a stationary or steady-state
scalar field.
Vector point function:
If corresponding to each point 𝑃(𝑥, 𝑦, 𝑧) of a region 𝑅 in space there corresponds a
unique vector function 𝑣 (𝑥, 𝑦, 𝑧) = 𝑣1𝑖̂ + 𝑣2 𝑗̂ + 𝑣3 𝑘̂ then, 𝑣 is called a vector point
function and R is called avector field.
For example,
(i) the velocity of a moving fluid at any instant.
(ii) The gravitational force.
(iii) The electric and magnetic field intensity.
Note: A vector field which is independent of time is called a stationary or steady-state
vector field.
Vector differential operator -
The vector differential operator is denoted by ∇ (del or nabla) and is defined as
, Faculty of Engineering and Technology.
Department of Applied Science & Humanities.
Gradient of a scalar field: - For a given scalar function ∅ (𝑥, 𝑦, 𝑧 ) the gradient of ∅ is
denoted by 𝑔𝑟𝑎𝑑 ∅ or ∇∅ is defined as
Example: Find the gradient of ∅ = 𝟑𝒙𝟐 𝒚 − 𝒚𝟑𝒛𝟐 at the point (𝟏, −𝟐, 𝟏).
Solution: We have,
At the point (1, −2,1)
∇ϕ= −12 𝑖̂ − 9 𝑗̂ − 16 𝑘̂.
2
Example: Evaluate 𝛻𝑒 (𝑟 ) , 𝒘𝒉𝒆𝒓𝒆 𝒓𝟐 = 𝒙𝟐 + 𝒚𝟐 + 𝒛𝟐
Solution:
𝑟 2 = 𝑥2 + 𝑦2 + 𝑧2
Differentiating 𝑟 partially with respect to 𝑥, 𝑦, 𝑧
𝜕𝑟 𝜕𝑟 𝑥
2𝑟 = 2𝑥 ⟹ =
𝜕𝑥 𝜕𝑥 𝑟
𝜕𝑟 𝜕𝑟 𝑦
2𝑟 = 2𝑦 ⟹ =
𝜕𝑦 𝜕𝑦 𝑟
𝜕𝑟 𝜕𝑟 𝑧
2𝑟 = 2𝑧 ⟹ =
𝜕𝑧 𝜕𝑧 𝑟
𝑟2 𝜕𝑒𝑟
2
𝜕𝑒𝑟
2 𝑟2 2
𝜕𝑒𝑟 𝜕𝑟
2
𝜕𝑒𝑟 𝜕𝑟
= 𝑖 𝜕𝑒𝜕𝑥 + 𝑗 = 𝑖 𝜕𝑒𝜕𝑟 𝜕𝑟
2
𝛻 𝑒𝑟 𝜕𝑦
+ 𝑘 𝜕𝑧 𝜕𝑥
+𝑗 𝜕𝑟 𝜕𝑦
+ 𝑘 𝜕𝑟 𝜕𝑧
2 𝑥 2 𝑦 2 𝑧 2
= 𝑖(2𝑟 𝑒 𝑟 ) + 𝑗 (2𝑟 𝑒 𝑟 ) + 𝑘 2𝑟 𝑒 𝑟 = 2𝑒 𝑟 (𝑥 𝑖̂ + 𝑦 𝑗̂ + 𝑧 𝑘̂ )
𝑟 𝑟 𝑟
, Faculty of Engineering and Technology.
Department of Applied Science & Humanities.
Example: Find a unit normal vector to the surface 𝒙𝟑 + 𝒚𝟑 + 𝟑𝒙𝒚𝒛 = 𝟑 at the
point (𝟏, 𝟐, −𝟏)
Solution:
At the point (1,2, −1)
Examples for Practice:
1. Find a unit normal vector to the surface 𝒙𝟐𝒚 + 𝟑𝒙𝒛𝟐 = 𝟖 at the point (𝟏, 𝟎, 𝟐)
2. Find the unit normal to the surface 𝒙𝟐 + 𝒙𝒚 + 𝒚𝟐 + 𝒙𝒚𝒛 at the point (𝟏, −𝟐, 𝟏)
Department of Applied Science & Humanities.
Subject: Mathematics –II (303191151)
Semester: 2nd Sem. B.Tech Programme (All Branches)
Lecture Notes: Unit – 5, Vector Calculus
Scalar point function:
• If corresponding to each point 𝑃(𝑥, 𝑦, 𝑧) of a region 𝑅 in space there
corresponds a unique scalar function 𝑓 = 𝑓 (𝑥, 𝑦, 𝑧) then, 𝑓 is called a scalar
point function and R is called a scalar field.
• For example,
(i) the temperature field in a body.
(ii) The pressure field of the air in the earth’s atmosphere.
(iii)The density of a body.
These quantities take different values at different points.
Note: A scalar field which is independent of time is called a stationary or steady-state
scalar field.
Vector point function:
If corresponding to each point 𝑃(𝑥, 𝑦, 𝑧) of a region 𝑅 in space there corresponds a
unique vector function 𝑣 (𝑥, 𝑦, 𝑧) = 𝑣1𝑖̂ + 𝑣2 𝑗̂ + 𝑣3 𝑘̂ then, 𝑣 is called a vector point
function and R is called avector field.
For example,
(i) the velocity of a moving fluid at any instant.
(ii) The gravitational force.
(iii) The electric and magnetic field intensity.
Note: A vector field which is independent of time is called a stationary or steady-state
vector field.
Vector differential operator -
The vector differential operator is denoted by ∇ (del or nabla) and is defined as
, Faculty of Engineering and Technology.
Department of Applied Science & Humanities.
Gradient of a scalar field: - For a given scalar function ∅ (𝑥, 𝑦, 𝑧 ) the gradient of ∅ is
denoted by 𝑔𝑟𝑎𝑑 ∅ or ∇∅ is defined as
Example: Find the gradient of ∅ = 𝟑𝒙𝟐 𝒚 − 𝒚𝟑𝒛𝟐 at the point (𝟏, −𝟐, 𝟏).
Solution: We have,
At the point (1, −2,1)
∇ϕ= −12 𝑖̂ − 9 𝑗̂ − 16 𝑘̂.
2
Example: Evaluate 𝛻𝑒 (𝑟 ) , 𝒘𝒉𝒆𝒓𝒆 𝒓𝟐 = 𝒙𝟐 + 𝒚𝟐 + 𝒛𝟐
Solution:
𝑟 2 = 𝑥2 + 𝑦2 + 𝑧2
Differentiating 𝑟 partially with respect to 𝑥, 𝑦, 𝑧
𝜕𝑟 𝜕𝑟 𝑥
2𝑟 = 2𝑥 ⟹ =
𝜕𝑥 𝜕𝑥 𝑟
𝜕𝑟 𝜕𝑟 𝑦
2𝑟 = 2𝑦 ⟹ =
𝜕𝑦 𝜕𝑦 𝑟
𝜕𝑟 𝜕𝑟 𝑧
2𝑟 = 2𝑧 ⟹ =
𝜕𝑧 𝜕𝑧 𝑟
𝑟2 𝜕𝑒𝑟
2
𝜕𝑒𝑟
2 𝑟2 2
𝜕𝑒𝑟 𝜕𝑟
2
𝜕𝑒𝑟 𝜕𝑟
= 𝑖 𝜕𝑒𝜕𝑥 + 𝑗 = 𝑖 𝜕𝑒𝜕𝑟 𝜕𝑟
2
𝛻 𝑒𝑟 𝜕𝑦
+ 𝑘 𝜕𝑧 𝜕𝑥
+𝑗 𝜕𝑟 𝜕𝑦
+ 𝑘 𝜕𝑟 𝜕𝑧
2 𝑥 2 𝑦 2 𝑧 2
= 𝑖(2𝑟 𝑒 𝑟 ) + 𝑗 (2𝑟 𝑒 𝑟 ) + 𝑘 2𝑟 𝑒 𝑟 = 2𝑒 𝑟 (𝑥 𝑖̂ + 𝑦 𝑗̂ + 𝑧 𝑘̂ )
𝑟 𝑟 𝑟
, Faculty of Engineering and Technology.
Department of Applied Science & Humanities.
Example: Find a unit normal vector to the surface 𝒙𝟑 + 𝒚𝟑 + 𝟑𝒙𝒚𝒛 = 𝟑 at the
point (𝟏, 𝟐, −𝟏)
Solution:
At the point (1,2, −1)
Examples for Practice:
1. Find a unit normal vector to the surface 𝒙𝟐𝒚 + 𝟑𝒙𝒛𝟐 = 𝟖 at the point (𝟏, 𝟎, 𝟐)
2. Find the unit normal to the surface 𝒙𝟐 + 𝒙𝒚 + 𝒚𝟐 + 𝒙𝒚𝒛 at the point (𝟏, −𝟐, 𝟏)
