Parul University

Faculty of Engineering & Technology

Department of Applied Sciences and Humanities

1st Year B.Tech Programme (All Branches)

Mathematics – II (303191151)

Unit – 5(a) Vector Calculus (Lecture Note)

Scalar point function:

• If to each point (𝑥, 𝑦, 𝑧) of a region 𝑅 in space there corresponds a number or a
scalar 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.

of a region 𝑅 in space there corresponds a vector

𝑣 (𝑥, 𝑦, 𝑧) = 𝑣1𝑖 + 𝑣2 𝑗 + 𝑣3 𝑘 then, 𝑣 is called a vector point function and R is called a
vector 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

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).
Sol:

At the point (1, −2,1)

𝛻∅ = −12 𝑖̂ − 9 𝑗̂ − 16 𝑘̂.

Example: Evaluate 𝜵𝒆𝒓𝟐, 𝒘𝒉𝒆𝒓𝒆 𝒓𝟐 = 𝒙𝟐 + 𝒚𝟐 + 𝒛𝟐

Solution:
𝑟2 = 𝑥2 + 𝑦2 + 𝑧2

Differentiating 𝑟 partially with respect to 𝑥, 𝑦, 𝑧

 𝜕𝑟 𝜕𝑟 𝑧
2𝑟 = 2𝑧 ⟹ =
𝜕𝑧 𝜕𝑧 𝑟

2 𝑥 2 𝑦 2 𝑧
= 𝑖 ( 2𝑟 𝑒 𝑟 ) + 𝑗 ( 2 𝑟 𝑒 𝑟 ) + 𝑘 2𝑟 𝑒 𝑟
𝑟 𝑟 𝑟

= 2𝑒𝑟2(𝑥 𝑖̂ + 𝑦 𝑗̂ + 𝑧 𝑘̂)

Example: Find a unit normal vector to the surface 𝒙𝟑 + 𝒚𝟑 + 𝟑𝒙𝒚𝒛 = 𝟑 at the

point (𝟏, 𝟐, −𝟏)
Sol.

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).
Directional Derivative: -

The directional derivative of scalar point function ∅(𝑥, 𝑦, 𝑧) in the direction of vector ,
is the component of in the direction of .
If is the unit vector in the direction of a, then the directional derivative of in the
direction of a is 𝐷∅ = ∇∅ ∗ 𝑎̂

Examples: Find the directional derivative of ∅(𝒙, 𝒚, 𝒛) = 𝒙𝟑 − 𝒙𝒚𝟐 − 𝒛 at point

̂
(1,1,0) in the direction of 𝒗 = 𝟐𝒊̂ − 𝟑𝒋̂ + 𝟔𝒌

Example: Find the directional derivative of ∅ = 𝟔𝒙𝟐 𝐲 + 𝟐𝟒𝒚𝟐 𝐳 − 𝟖𝒛𝟐 𝒙 at (1,1,1)

̂ . Hence, find the maximum value.
in the direction of 𝒗 = 𝟐𝒊̂ − 𝟐𝒋̂ + 𝒌
 𝛻𝜙 = (12𝑥𝑦 − 8𝑧2)𝑖 + (6𝑥2 + 48𝑦𝑧)𝑗 + (24𝑦2 − 16𝑧𝑥)𝑘

𝛻𝜙1,1,1 = 4𝑖 + 54𝑗 + 8𝑘

Directional derivative in the direction of 𝑣 = (2𝑖 − 2𝑗 + 𝑘) at the point (1,1,1)

Maximum value of directional derivative = |𝛻𝜙|

Example for Practice

1. Find the directional derivative of 𝝋(𝒙, 𝒚, 𝒛) = 𝒙𝒚𝟐 + 𝒚𝒛𝟑 at the point P(𝟐, −𝟏,
𝟏) in the direction of PQ where Q is the point (𝟑, 𝟏, 𝟑)
2. In what direction from (−𝟏, 𝟏, 𝟐) is the directional derivative of 𝝋 = 𝒙𝒚𝟐 𝒛𝟑 a
maximum? Find also the magnitude of this maximum.
3. Find the directional derivative of the scalar function 𝝋 = 𝒙𝒚𝒛 in the direction
of the outer normal to the surface 𝒛 = 𝒙𝒚 at the point (𝟑, 𝟏, 𝟑).
4. Find the directional derivative of 𝝋 = 𝒙𝒚 + 𝒚𝒛 + 𝒛𝒙 at (𝟏, 𝟐, 𝟎) in the direction
̂.
of 𝒊̂ + 𝟐𝒋̂ + 𝟐𝒌

Divergence of a vector function:

Let 𝐹 = 𝐹1𝑖̂ + 𝐹2 𝑗̂ + 𝐹3𝑘̂ be a vector function then, divergence of F is
𝜕 𝜕 𝜕 𝜕𝐹 𝜕𝐹 𝜕𝐹
div F OR ∇𝐹 = (𝑖̂ + 𝑗̂ + 𝑘̂ ) (𝐹1 𝑖̂ + 𝐹2 𝑗̂ + 𝐹3 𝑘̂) = 1 + 2 + 3
𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑥 𝜕𝑦 𝜕𝑧
 Note:
1. If 𝛻. 𝐹 = 0 then, the vector function F is called solenoidal or incompressible
2. In hydrodynamics (the study of fluid motion), a velocity field that is divergence
free is called incompressible.
3. In the study of electricity and magnetism, a vector field that is divergence free is
called solenoidal.

Example: If 𝑭 = 𝒙𝟐 𝒛 𝒊̂ − 𝟐𝒚𝟑 𝒛𝟑 𝒋̂ + 𝒙𝒚𝟐𝒛 𝒌̂ then, find divergence of F at (𝟏, −𝟏, 𝟏)

Sol. Here, 𝐹 = 𝑥2 𝑧 𝑖̂ − 2𝑦3 𝑧3 𝑗̂ + 𝑥𝑦2𝑧 𝑘̂

At ( 1,−1,1)
𝛻𝐹 = −3

Example: Show that 𝐀 = 𝟑𝒚𝟒𝒛𝟐𝒊 + 𝟒𝒙𝟑𝒛𝟐𝒋 − 𝟑𝒙𝟐𝒚𝟐𝒌 is a solenoidal.

Solution:

Hence Given function is solenoidal.

Example for Practice:
1. Determine the constant a such that 𝑨 = (𝒂𝒙𝟐 𝒚 + 𝒚𝒛)𝒊̂ + (𝒙𝒚𝟐 + 𝒙𝒛𝟐 )𝒋̂ +
(𝟐𝒙𝒚𝒛 − 𝟐𝒙𝟐 𝒚𝟐 )𝒌 ̂ is solenoidal.
2. Find div 𝐅⃗, where 𝐅⃗ = grad(𝒙3+𝒚3+𝒛3−3𝒙𝒚𝒛)
̂ find 𝛁. 𝑭 at a point (1,-1,1)
⃗⃗ = 𝒙𝒚𝟐 𝒊̂ + 𝟐𝒙𝟐 𝒚𝒛𝒋̂ − 𝟑𝒚𝒛𝟐 𝒌
3. If 𝑭
⃗⃗ = (𝒙𝟐 − 𝒚𝟐 + 𝟐𝒙𝒛)𝒊̂ + (𝒙𝒛 − 𝒙𝒚 + 𝒚𝒛)𝒋̂ − (𝒛𝟐 + 𝒙𝟐 )𝒌
4. If 𝑭 ̂ then find 𝛁. 𝑭 .

Curl
Let 𝐹= 𝐹1𝑖̂ + 𝐹2 𝑗̂ + 𝐹3𝑘̂ be a vector function then, curl of F is

Note: - If 𝛻 × 𝐹 = 0 then, the vector function F is called Irrotational or conservative.

Example: If 𝑭 = 𝒙𝒛𝟑 𝒊̂ − 𝟐𝒙𝟐 𝒚𝒛𝒋̂ + 𝟐𝒚𝒛𝟒 𝒌 ̂ then, find curl of F at (1,-1,1)

Sol. Here, 𝐹 = 𝑥𝑧 3 𝑖̂ − 2𝑥 2 𝑦𝑧𝑗̂ + 2𝑦𝑧 4 𝑘̂

At point (1, −1,1)

.
 ̂ is Irrotational.
Example: Show that 𝒓 = 𝒙𝒊̂ + 𝒚𝒋̂ + 𝒛𝒌

Solution:

Therefore, 𝑟 = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧𝑘̂ is Irrotational

̂) at a point(1,2,3)
Example: Find curl of 𝑨 = 𝒆𝒙𝒚𝒛 (𝒊̂ + 𝒋̂ + 𝒌
Solution:

At

Example for Practice:

̂ find 𝛁 × 𝑭 at a point (1,-1,1)
⃗⃗ = 𝒙𝒚𝟐 𝒊̂ + 𝟐𝒙𝟐 𝒚𝒛𝒋̂ − 𝟑𝒚𝒛𝟐 𝒌
1. If 𝑭
⃗⃗ = (𝒙𝟐 − 𝒚𝟐 + 𝟐𝒙𝒛)𝒊̂ + (𝒙𝒛 − 𝒙𝒚 + 𝒚𝒛)𝒋̂ − (𝒛𝟐 + 𝒙𝟐 )𝒌
2. If 𝑭 ̂ then find 𝛁 × 𝑭 .
3. Find div(grad 𝜑) and curl(grad 𝜑) at (1,1,1) for 𝜑 = 𝐱𝟐𝐲𝟑𝐳𝟒