This document discusses vector analysis and defines solenoidal and irrotational vector functions. It provides examples to verify whether specific vector functions are solenoidal or irrotational. Specifically: It defines key concepts in vector analysis including vectors, vector operations, and vector-valued functions. It then discusses the history and development of vector analysis. It defines a solenoidal (divergence-free) vector function as one where the divergence is equal to zero. An example verifies that a given vector function is not solenoidal. It defines an irrotational (curl-free) vector function as one where the curl is equal to zero. An example verifies that a given vector function is irrotational

1. Green university of Bangladesh
Md. Al-Amin ID: 172015031
Shakiuzzamn ID: 172015027
Mahabubur Rahim ID: 172015040
Topic: Verification of Solenoidal & Irrotational
Department of CSE
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2. Vector Analysis
Vector: A vector is a quantity or phenomenon that has two independent
properties: magnitude and direction. Examples of vector quantities
displacement, velocity, acceleration, force, etc.
Vector analysis uses, applies and extends the methods of differential and integral
calculus to vectors and vector valued functions.
The Dot product, Cross product, Scalar multiplication, Gradient, Divergence, Curl,
Directional derivative, Stokes' theorem, Green's theorem, the Divergence theorem,
and other mathematical concepts and notions related to vectors are studied within
the framework of vector analysis .
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3. History
When and how did vector analysis arise and develop?
Vector analysis arose only in the period after 1831, in the 19th century
when Josiah Willard Gibbs and Oliver Heaviside independently developed
vector analysis to express the new laws of electromagnetism discovered by
the Scottish physicist James Clerk Maxwell. Now three earlier developments
deserve attention as leading up to it. These three developments are-
 The discovery and geometrical representation of complex numbers.
 Leibniz’s search for a geometry of position.
 The idea of a parallelogram of forces or velocities.
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4. Solenoidal
A vector function 𝑓 is said to Solenoidal on divergence free. That
means if div 𝑓 = 0.
Divergence: If v = 𝑣1 𝑖^
+ 𝑣2 𝑗^
+ 𝑣3 𝑘^
is define and differentiable at
each point (x,y,z). The divergence of v is define as
div v = ∇.v
=
𝜕 𝑣1
𝜕𝑥
+
𝜕 𝑣2
𝜕𝑦
+
𝜕 𝑣3
𝜕𝑧
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5. Verification
Verifay that, 𝑓 =2𝑥2 𝑧 𝑖^ − 𝑥𝑦2 𝑧 − 3𝑦𝑧2 𝑘^ is Solenoidal or not?
Solution:
we know-
div 𝑓 = ∇. 𝑓
=𝑖^ 𝜕 𝑣1
𝜕𝑥
+ 𝑗^ 𝜕 𝑣2
𝜕𝑦
+ 𝑘^ 𝜕 𝑣3
𝜕𝑧
=
𝜕
𝜕𝑥
(2𝑥2
𝑧 ) +
𝜕
𝜕𝑦
(−𝑥𝑦2
𝑧) +
𝜕
𝜕𝑧
(−3𝑦𝑧2
)
= 4𝑥𝑧 − 2𝑥𝑦𝑧 − 6𝑦𝑧
So f ≠ 0
So f is not Solenoidal.
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6. Irrotational
A vector function 𝑓 is said to irrotational on curl free. If curl 𝑓 = o.
Curl: If v = 𝑣1 𝑖^
+ 𝑣2 𝑗^
+ 𝑣3 𝑘^
is define and differentiable at each
point (x,y,z). The curl of v is define as-
Curl v = ∇ × v
=
𝑖^
𝑗^
𝑘^
𝜕
𝜕𝑥
𝜕
𝜕𝑦
𝜕
𝜕𝑧
𝑣1 𝑣2 𝑣3
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7. Verification
Verifay that, 𝑓 = 𝑖^
6𝑥𝑦 + 𝑧2
+ 𝑗^
3𝑥2
+ 𝑧 + 𝑘^
(3𝑥𝑧2
− 𝑦) is irrotational or not?
Solution:
We know-
curl f = ∇ × 𝑓
=
𝑖^
𝑗^
𝑘^
𝜕
𝜕𝑥
𝜕
𝜕𝑦
𝜕
𝜕𝑧
6𝑥𝑦 + 𝑧2
3𝑥2
+ 𝑧 3𝑥𝑧2
− 𝑦
= 𝑖^
{
𝜕
𝜕𝑦
(3𝑥𝑧2
− 𝑦) −
𝜕
𝜕𝑧
(3𝑥2
+ 𝑧)} − 𝑗^
{
𝜕
𝜕𝑥
(3𝑥𝑧2
− 𝑦) –
𝜕
𝜕𝑧
(6𝑥𝑦 + 𝑧2
)} + 𝑘^
{
𝜕
𝜕𝑥
(3𝑥2
+ 𝑧) −
𝜕
𝜕𝑦
(6𝑥𝑦 + 𝑧2
)}
= 𝑖^(− 1+1) − 𝑗^(3𝑧2 − 3𝑧2) + 𝑘^(6𝑥 − 6𝑥)
= 0 So f is irrotational.
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