19MAT111
Multivariable Optimization
for 2nd semester ECE, EEE
-Dr. Sarada Jayan
4/26/2020
Sarada Jayan
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Vector Differentiation (Review)
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Scalar functions, Vector functions
Applications of partial derivatives
Vector plots
Parametric representation of curves, C:തr(t)
Velocity and Acceleration of moving particles
Vector Differentiation
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4/26/2020
Gradient of a scalar function, ∇𝑓 (vector)
Divergence of a vector function, ∇ ∙ 𝑣ҧ (scalar)
Curl of a vector function, ∇ × 𝑣ҧ (vector)
Laplacian of a scalar function ∇2 𝑓 (scalar)
Applications of grad, div and curl
Solenoidal/Incompressible fields, Irrotational/ Conservative fields
Sarada Jayan
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Scalar Potential
• If a vector field 𝑣ҧ is irrotational i.e. if curl 𝑣ҧ = 0,
there will always exist a scalar function f such that :
𝑣ҧ = 𝑔𝑟𝑎𝑑 𝑓 = ∇𝑓
=
𝜕𝑓
𝑖Ƹ
𝜕𝑥
+
𝜕𝑓
𝑗Ƹ
𝜕𝑦
+
𝜕𝑓 ෠
𝑘
𝜕𝑧
This function f is called the scalar potential of 𝑣.ҧ
Examples:
෠
1. If 𝑣ҧ = 2𝑥 𝑖Ƹ + 2y𝑗Ƹ + 2𝑧𝑘,
then the scalar potential is 𝑓 = 𝑥 2 + 𝑦 2 + 𝑧 2 + 𝑐
෠
2. If 𝑣ҧ = 𝑦𝑧𝑖Ƹ + xz𝑗Ƹ + 𝑥𝑦𝑘,
then the scalar potential is 𝑓 = 𝑥𝑦𝑧 + 𝑐
4/26/2020
Sarada Jayan
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How to find the scalar Potential … Problems
1. Find the scalar potential for the following vector functions, if it exists.
a. 𝑣ҧ = 3𝑥 2 𝑖Ƹ + 3𝑦 2 𝑗Ƹ + 3𝑧 2 𝑘෠
Solution:
curl 𝑣ҧ =
𝑖Ƹ
𝑗Ƹ
𝑘෠
𝜕
𝜕𝑥
3𝑥 2
𝜕
𝜕𝑦
3𝑦 2
𝜕
𝜕𝑧
3𝑧 2
i.e., 𝑣ҧ =
𝜕𝑓
𝑖Ƹ
𝜕𝑥
+
𝜕𝑓
𝑗Ƹ
𝜕𝑦
⇒
𝜕𝑓
𝜕𝑥
= 3𝑥 2 ;
+
𝜕𝑓
𝑘෠
𝜕𝑧
= [0,0,0] ⇒ 𝑣ҧ is irrotational and has a potential, f.
= 3𝑥 2 𝑖Ƹ + 3𝑦 2 𝑗Ƹ + 3𝑧 2 𝑘෠
𝜕𝑓
𝜕𝑦
= 3𝑦 2 ;
𝜕𝑓
𝜕𝑧
= 3𝑧 2 ;
𝜕𝑓
𝜕𝑥
𝜕𝑓
𝜕𝑦
= 3𝑥 2 ⇒ 𝑓 = 𝑥 3 + 𝑓1 𝑦, 𝑧 ----- (1) [Integration w.r.t. x]
𝜕𝑓
𝜕𝑧
= 3𝑧 2 ⇒ 𝑓 = 𝑧 3 + 𝑓3 𝑥, 𝑦 ----- (3) [Integration w.r.t. z]
= 3𝑦 2 ⇒ 𝑓 = 𝑦 3 + 𝑓2 𝑥, 𝑧 ----- (2) [Integration w.r.t. y]
From (1), (2) and (3), we can get 𝒇 = 𝒙𝟑 + 𝒚𝟑 + 𝒛𝟑 + 𝒄
4/26/2020
Sarada Jayan
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How to find the scalar Potential… Problems
1. Find the scalar potential for the following vector functions, if it exists.
b. 𝑣ҧ = 6𝑥 2 𝑦𝑧 2 𝑖Ƹ + 2𝑥 3 𝑧 2 𝑗Ƹ + 4𝑥 3 𝑦𝑧 𝑘෠
Solution:
curl 𝑣ҧ =
𝑖Ƹ
𝑗Ƹ
𝑘෠
𝜕
𝜕𝑥
6𝑥 2 𝑦𝑧 2
𝜕
𝜕𝑦
2𝑥 3 𝑧 2
𝜕
𝜕𝑧
4𝑥 3 𝑦𝑧
= [4𝑥 3 𝑧 − 4𝑥 3 𝑧, −12𝑥 2 𝑦𝑧 + 12𝑥 2 𝑦𝑧, 6𝑥 2 𝑧 2 − 6𝑥 2 𝑧 2 ]
= [0,0,0] ⇒ 𝑣ҧ is irrotational and has a potential, f.
i.e., 𝑣ҧ =
𝜕𝑓
𝑖Ƹ
𝜕𝑥
+
𝜕𝑓
𝑗Ƹ
𝜕𝑦
⇒
𝜕𝑓
𝜕𝑥
= 6𝑥 2 𝑦𝑧 2 ;
+
𝜕𝑓 ෠
𝑘
𝜕𝑧
= 6𝑥 2 𝑦𝑧 2 𝑖Ƹ + 2𝑥 3 𝑧 2 𝑗Ƹ + 4𝑥 3 𝑦𝑧 𝑘෠
𝜕𝑓
𝜕𝑦
= 2𝑥 3 𝑧 2 ;
𝜕𝑓
𝜕𝑧
= 4𝑥 3 𝑦𝑧;
𝜕𝑓
𝜕𝑥
𝜕𝑓
𝜕𝑦
= 6𝑥 2 𝑦𝑧 2 ⇒ 𝑓 = 2𝑥 3 𝑦𝑧 2 + 𝑓1 𝑦, 𝑧 ----- (1) [Integration w.r.t. x]
𝜕𝑓
𝜕𝑧
= 4𝑥 3 𝑦𝑧⇒ 𝑓 = 2𝑥 3 𝑦𝑧 2 + 𝑓3 𝑥, 𝑦 ----- (3) [Integration w.r.t. z]
= 2𝑥 3 𝑧 2 ⇒ 𝑓 = 2𝑥 3 𝑦𝑧 2 + 𝑓2 𝑥, 𝑧 ----- (2) [Integration w.r.t. y]
From (1), (2) and (3), we can get 𝒇 = 𝟐𝒙𝟑 𝒚𝒛𝟐 + 𝒄
4/26/2020
Sarada Jayan
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How to find the scalar Potential…Problems
1. Find the scalar potential for the following vector functions, if it exists.
c. 𝑣ҧ = 𝑥𝑦𝑖Ƹ + 2𝑥𝑦𝑗Ƹ + 3𝑦𝑧 𝑘෠
Solution:
curl 𝑣ҧ =
𝑖Ƹ
𝑗Ƹ
𝑘෠
𝜕
𝜕𝑥
𝜕
𝜕𝑦
𝜕
𝜕𝑧
= [3𝑧, 0, 2𝑦 − 𝑥] ≠ [0,0,0]
𝑥𝑦 2𝑥𝑦 3𝑦𝑧
⇒ 𝑣ҧ is not irrotational and so does not have a potential.
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Sarada Jayan
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How to find the scalar Potential…Problems
2. Find the scalar potential for the following irrotational function:
𝑣ҧ = 𝑦 sin 𝑧 − sin 𝑥 𝑖Ƹ + 𝑥 sin 𝑧 + 2𝑦𝑧 𝑗Ƹ + 𝑥𝑦 cos 𝑧 + 𝑦 2 𝑘෠
Solution:
𝜕𝑓
𝜕𝑓
𝜕𝑓
𝑣ҧ =
𝑖Ƹ +
𝑗Ƹ +
𝑘෠
𝜕𝑥
𝜕𝑦
𝜕𝑧
= 𝑦 sin 𝑧 − sin 𝑥 𝑖Ƹ + 𝑥 sin 𝑧 + 2𝑦𝑧 𝑗Ƹ + 𝑥𝑦 cos 𝑧 + 𝑦 2 𝑘෠
𝜕𝑓
𝜕𝑥
⇒
𝜕𝑓
𝜕𝑥
𝜕𝑓
𝜕𝑦
𝜕𝑓
𝜕𝑧
= 𝑦 sin 𝑧 − sin 𝑥;
𝜕𝑓
𝜕𝑦
= 𝑦 sin 𝑧 − sin 𝑥;
𝜕𝑓
𝜕𝑧
= 𝑥𝑦 cos 𝑧 + 𝑦 2 ;
= 𝑦 sin 𝑧 − sin 𝑥 ⇒ 𝑓 = 𝑥𝑦 sin 𝑧 + cos 𝑥 + 𝑓1 𝑦, 𝑧 ----- (1)
[Integration w.r.t. x]
= 𝑥 sin 𝑧 + 2𝑦𝑧 ⇒ 𝑓 = 𝑥𝑦 sin 𝑧 + 𝑦 2 𝑧 + 𝑓2 𝑥, 𝑧 ----- (2)
[Integration w.r.t. y]
= 𝑥𝑦 cos 𝑧 + 𝑦 2 ⇒ 𝑓 = 𝑥𝑦 sin 𝑧 + 𝑦 2 𝑧 + 𝑓3 𝑥, 𝑦 ----- (3)
[Integration w.r.t. z]
From (1), (2) and (3), we can get 𝒇 = 𝒙𝒚 𝒔𝒊𝒏 𝒛 + 𝒄𝒐𝒔 𝒙 + 𝒚𝟐 𝒛 + 𝒄
4/26/2020
Sarada Jayan
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How to find the scalar Potential…Problems
3. Find the scalar potential f for the following irrotational function 𝑣ҧ given that
𝑓 1,1,1 = 14 and 𝑣ҧ = 𝑦 2 − 2𝑥𝑦𝑧 3 𝑖Ƹ + 3 + 2𝑥𝑦 − 𝑥 2 𝑧 3 𝑗Ƹ + 8𝑧 3 − 3𝑥 2 𝑦𝑧 2 𝑘෠
Solution:
𝜕𝑓
𝜕𝑓
𝜕𝑓
𝑣ҧ =
𝑖Ƹ +
𝑗Ƹ +
𝑘෠
𝜕𝑥
𝜕𝑦
𝜕𝑧
= 𝑦 2 − 2𝑥𝑦𝑧 3 𝑖Ƹ + 3 + 2𝑥𝑦 − 𝑥 2 𝑧 3 𝑗Ƹ + 8𝑧 3 − 3𝑥 2 𝑦𝑧 2 𝑘෠
𝜕𝑓
𝜕𝑥
⇒
= 𝑦 2 − 2𝑥𝑦𝑧 3 ;
𝜕𝑓
𝜕𝑦
= 3 + 2𝑥 − 𝑥 2 𝑧 3 ;
𝜕𝑓
𝜕𝑧
= 8𝑧 3 − 3𝑥 2 𝑦𝑧 2 ;
𝜕𝑓
𝜕𝑥
𝜕𝑓
𝜕𝑦
= 𝑦 2 − 2𝑥𝑦𝑧 3 ⇒ 𝑓 = 𝑥𝑦 2 − 𝑥 2 𝑦𝑧 3 + 𝑓1 𝑦, 𝑧 ----- (1)
𝜕𝑓
𝜕𝑧
= 8𝑧 3 − 3𝑥 2 𝑦𝑧 2 ⇒ 𝑓 = 2𝑧 4 − 𝑥 2 𝑦𝑧 3 + 𝑓3 𝑥, 𝑦 ----- (3)
[Integration w.r.t. x]
= 3 + 2𝑥𝑦 − 𝑥 2 𝑧 3 ⇒ 𝑓 = 3𝑦 + 𝑥𝑦 2 − 𝑥 2 𝑦𝑧 3 + 𝑓2 𝑥, 𝑧 ---- (2)[Integration w.r.t. y]
[Integration w.r.t. z]
From (1), (2) and (3), we can get 𝒇 = 𝒙𝒚𝟐 − 𝒙𝟐 𝒚𝒛𝟑 + 𝟑𝒚 + 𝟐𝒛𝟒 + 𝒄
𝑓 1,1,1 = 14 ⇒ 14=5+c ⇒ c=9
⇒ 𝒇 = 𝒙𝒚𝟐 − 𝒙𝟐 𝒚𝒛𝟑 + 𝟑𝒚 + 𝟐𝒛𝟒 + 𝟗
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Sarada Jayan
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Exercise
4. Find the scalar potential for the following irrotational functions:
a. 𝑣ҧ = 𝑦𝑧 + 2𝑥 𝑖Ƹ + 𝑥𝑧 − 𝑧 𝑗Ƹ + 𝑥𝑦 − 𝑦 𝑘෠
b. 𝑣ҧ =
𝑥
𝑖Ƹ
𝑥 2 +𝑦 2
+
𝑦
𝑗Ƹ
𝑥 2 +𝑦 2
1
c. 𝑣ҧ = 3𝑒 𝑥 𝑖Ƹ + 6 cos 𝑦 𝑗Ƹ + 𝑘෠
𝑧
𝑦
𝑧
𝑥
𝑧
d. 𝑣ҧ = 𝑖Ƹ + 𝑗Ƹ +
−𝑥𝑦 ෠
𝑘
𝑧2
5. Check if the scalar potential exists and if it exists find it.
a. 𝑣ҧ = 𝑦𝑒 𝑥 𝑖Ƹ + 𝑒 𝑥 𝑗Ƹ + 𝑘෠
b. 𝑣ҧ = 𝑥𝑦𝑖Ƹ + 2𝑥𝑦𝑗Ƹ
c. 𝑣ҧ = 2𝑥𝑧𝑖Ƹ + 𝑗Ƹ + (𝑥 2 + 𝑦 2 ) 𝑘෠
d. 𝑣ҧ = 2𝑥𝑖Ƹ − 2y𝑗Ƹ + (3𝑧 2 − 2𝑧) 𝑘෠
4/26/2020
Sarada Jayan
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Exercise
4. Find the scalar potential for the following irrotational functions:
a. 𝑣ҧ = 𝑦𝑧 + 2𝑥 𝑖Ƹ + 𝑥𝑧 − 𝑧 𝑗Ƹ + 𝑥𝑦 − 𝑦 𝑘෠
Solution: 𝑓 = 𝑥𝑦𝑧 + 𝑥 2 − 𝑦𝑧 + 𝑐
𝑥
𝑦
b. 𝑣ҧ = 𝑥 2+𝑦2 𝑖Ƹ + 𝑥 2+𝑦2 𝑗Ƹ
1
Solution: 𝑓 = 2 ln 𝑥 2 + 𝑦 2 + 𝑐
1
c. 𝑣ҧ = 3𝑒 𝑥 𝑖Ƹ + 6 cos 𝑦 𝑗Ƹ + 𝑧 𝑘෠
Solution: 𝑓 = 3𝑒 𝑥 + 6 sin 𝑦 + ln 𝑧 + 𝑐
𝑦
𝑥
−𝑥𝑦
d. 𝑣ҧ = 𝑧 𝑖Ƹ + 𝑧 𝑗Ƹ + 𝑧 2 𝑘෠
Solution: 𝑓 =
𝑥𝑦
𝑧
+𝑐
5. Check if the scalar potential exists and if it exists find it.
a. 𝑣ҧ = 𝑦𝑒 𝑥 𝑖Ƹ + 𝑒 𝑥 𝑗Ƹ + 𝑘෠
Solution: Curl 𝑣ҧ =0 and potential is 𝑓 = 𝑦𝑒 𝑥 + 𝑧 + 𝑐
b. 𝑣ҧ = 𝑥𝑦𝑖Ƹ + 2𝑥𝑦𝑗Ƹ
Solution: Curl 𝑣ҧ ≠0 and so potential does not exists
c. 𝑣ҧ = 2𝑥𝑧𝑖Ƹ + 𝑗Ƹ + (𝑥 2 + 𝑦 2 ) 𝑘෠
Solution: Curl 𝑣ҧ ≠ 0 and so potential does not exists
d. 𝑣ҧ = 2𝑥 𝑖Ƹ − 2y𝑗Ƹ + (3𝑧 2 − 2𝑧) 𝑘෠
Solution: Curl 𝑣ҧ = 0 and potential is 𝑓 = 𝑥 2 −𝑦 2 +𝑧 3 − 𝑧 2 + 𝑐
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Sarada Jayan
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