GATE Engineering Maths : Vector Calculus

Section 2 : Calculus
Topic 7 : Vector Calculus
Vector Basics
 Position vector :- The position vector of the point P(x, y, z) in the space is
𝑟 = 𝑥 𝑖 + 𝑦 𝑗 + 𝑧 𝑘
𝑟 = 𝑥2 + 𝑦2 + 𝑧2
 In parametric form, 𝑟 = 𝑥 𝑡 𝑖 + 𝑦 𝑡 𝑗 + 𝑧(𝑡) 𝑘
 Let, 𝑎 = 𝑎1 𝑖 + 𝑎2 𝑗 + 𝑎3 𝑘, 𝑏 = 𝑏1 𝑖 + 𝑏2 𝑗 + 𝑏3 𝑘
𝑎.𝑏 = 𝑎 𝑏 cos 𝑎. 𝑏 = 𝑎1 𝑏1 + 𝑎2 𝑏2 + 𝑎3 𝑏3
𝑎 × 𝑏 = 𝑎 |𝑏| sin( 𝑎. 𝑏) 𝑛 , where n is vector of unit length perpendicular to the plane
containing 𝑎 & 𝑏.
 𝑎 × 𝑏 =
𝑖 𝑗 𝑘
𝑎1 𝑎2 𝑎3
𝑏1 𝑏2 𝑏3
Vector Basics
 Area of ∆𝑂𝐴𝐵 =
1
2
𝑂𝐴 × 𝑂𝐵 =
1
2
𝑎 × 𝑏
 Area of ∆𝐴𝐵𝐶 =
1
2
𝐴𝐵 × 𝐴𝐶 =
1
2
(𝑏 − 𝑎) × ( 𝑐 − 𝑎)
 Area of parallelogram = | 𝑎 × 𝑏|
 Scalar triple product :- 𝑎 × 𝑏 . 𝑐 = 𝑎. 𝑏 × 𝑐 = [ 𝑎 𝑏 𝑐] =
𝑎1 𝑎2 𝑎3
𝑏1 𝑏2 𝑏3
𝑐1 𝑐2 𝑐3
 Vector triple product :- 𝑎 × 𝑏 × 𝑐 = 𝑐. 𝑎 𝑏 − 𝑏. 𝑎 𝑐
Vector Differentiation
 Let 𝑟 𝑡 = 𝑓(𝑡) then,
𝑑 𝑟
𝑑𝑡
= lim
∆𝑡→0
𝑓 𝑡+∆𝑡 − 𝑓(𝑡)
∆𝑡
 If t is a time variable then
𝑑 𝑟
𝑑𝑡
represents a velocity vector.
1.
𝑑 𝑟
𝑑𝑡
is a vector in direction of tangent to the curve at that point.
2. If 𝑓(𝑡) is constant in magnitude then 𝐹.
𝑑 𝐹
𝑑𝑡
= 0
3. If 𝑓(𝑡) has constant direction then, 𝐹 ×
𝑑 𝐹
𝑑𝑡
= 0
Vector Differentiation
 Vector differential operator :- 𝛻 (nebla)
𝛻 = 𝑖
𝜕
𝜕𝑥
+ 𝑗
𝜕
𝜕𝑦
+ 𝑘
𝜕
𝜕𝑧
 Gradient of a scalar function :- Let 𝜑(𝑥, 𝑦, 𝑧) be a differentiable scalar point function then
gradient of scalar is denoted by grad 𝜑 or 𝛻𝜑 = 𝑖
𝜕𝜑
𝜕𝑥
+ 𝑗
𝜕𝜑
𝜕𝑦
+ 𝑘
𝜕𝜑
𝜕𝑧
 Where, 𝛻𝜑 is vector normal to surface 𝜑.
 Unit vector normal to surface 𝜑 can be given as
𝛻𝜑
|𝛻𝜑|
.
Vector Differentiation
 Directional derivative :- The directional derivative of differentiable scalar function 𝜑(𝑥, 𝑦, 𝑧)
in the direction of 𝑎 is given by, 𝛻𝜑.
𝑎
|𝑎|
 Let 𝑎 = 𝑖, then,
 D.D. = 𝛻𝜑.
𝐼
| 𝐼|
= ( 𝑖
𝜕𝜑
𝜕𝑥
+ 𝑗
𝜕𝜑
𝜕𝑦
+ 𝑘
𝜕𝜑
𝜕𝑧
). 𝑖
=
𝜕𝜑
𝜕𝑥
 Angle between surfaces :- It is the angle between the normal to the surfaces at the point of
intersection. Let 𝜃 be the angle between the surfaces 𝜑1 𝑥, 𝑦, 𝑧 = 𝐶1 & 𝜑2 𝑥, 𝑦, 𝑧 = 𝐶2
then,
cos 𝜃 =
𝛻∅1 𝛻𝜑2
|𝛻∅1| |𝛻𝜑2|
Vector Differentiation
 Divergence of a vector function :- Let 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be a differential vector
point function then,
𝑑𝑖𝑣 𝐹 = 𝛻. 𝐹 =
𝜕𝐹1
𝜕𝑥
+
𝜕𝐹2
𝜕𝑦
+
𝜕𝐹3
𝜕𝑧
Note :- If 𝛻. 𝐹 = 0 then 𝐹 is called solenoidal vector.
 Curl of a vector function :- 𝑐𝑢𝑟𝑙 𝐹 = 𝛻 × 𝐹 =
𝑖 𝑗 𝑘
𝜕
𝜕𝑥
𝜕
𝜕𝑦
𝜕
𝜕𝑧
𝐹1 𝐹2 𝐹3
Note :- If 𝛻 × 𝐹 = 0 then 𝐹 is called irrotational vector.
If 𝑣 = velocity vector and 𝑤 = angular velocity, 𝑤 =
1
2
𝑐𝑢𝑟𝑙 𝑣
Vector Differentiation
 Scalar Potential Function :- If for every rotational vector, a scalar function 𝜑 exist such that
𝐹 = 𝛻𝜑, then 𝜑 is said to be scalar potential function.
 Note :-
1) 𝑐𝑢𝑟𝑙 𝑔𝑟𝑎𝑑 𝜑 = 0
2) 𝑑𝑖𝑣 𝑐𝑢𝑟𝑙 𝐹 = 0
3) 𝑑𝑖𝑣 𝑔𝑟𝑎𝑑 𝜑 = 𝛻 𝛻𝜑 = 𝛻2 𝜑, where 𝛻2=
𝜕2
𝜕𝑥2 +
𝜕2
𝜕𝑦2 +
𝜕2
𝜕𝑧2 (𝛻2 Laplacian Operator)
Vector Integration
 Line integral :- An integral evaluated over a curve is called line integral.
 Let, 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘, be a differentiable point function defined at each point on
curve ‘c’ then its line integral is
𝑐
𝐹. 𝑑𝑟 =
𝑐
𝐹1 𝑑𝑥 + 𝐹2 𝑑𝑦 + 𝐹3 𝑑𝑧
 If ‘c’ is closed curve  𝑐
𝐹. 𝑑𝑟
Note :- If 𝐹 is irrotational then, the line integral of 𝐹 is independent of path.
When, 𝐹 is irrotational  𝑎
𝑏
𝐹. 𝑑𝑟 = 𝜑 𝑏 − 𝜑 𝑎 (Where, 𝜑 is scalar potential function)
Vector Integration
 Green’s theorem :- Let, M(x, y) & N(x, y) be continuous function having continuous first
order partial derivative defined in the closed region R bounded by closed curve ‘c’ then,
𝑐
(𝑀𝑑𝑥 + 𝑁𝑑𝑦) =
𝑅
𝜕𝑁
𝜕𝑥
−
𝜕𝑀
𝜕𝑦
𝑑𝑥 𝑑𝑦
 Surface Integral :- Let 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be a differentiable vector point function
defined over the surface S then, its surface integration is
S
𝐹. d 𝑠 =
S
𝐹. 𝑛 ds
Where, 𝑛  unit outward drawn normal to the surface
Vector Integration
 Methods of evaluation of surface integral:-
1. If 𝑅1 is the projection of ‘S’ on to x-y plane then, 𝑠
𝐹 . 𝑛 𝑑𝑠 = 𝑅1
𝐹 . 𝑛
𝑑𝑥 𝑑𝑦
|𝑛 𝑘|
2. If 𝑅2 is the projection of ‘S’ on to y-z plane then, 𝑠
𝐹 . 𝑛 𝑑𝑠 = 𝑅2
𝐹 . 𝑛
𝑑𝑦 𝑑𝑧
|𝑛 𝑖|
3. If 𝑅3 is the projection of ‘S’ on to x-z plane then, 𝑠
𝐹 . 𝑛 𝑑𝑠 = 𝑅3
𝐹 . 𝑛
𝑑𝑧 𝑑𝑥
|𝑛 𝑗|
Vector Integration
 Let, 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be the differential vector point function defined in volume V,
then its volume integral is 𝑉
𝐹 𝑑𝑣
 Gauss Divergence Theorem :- Let s be a closed surface enclosing a volume V & 𝐹 𝑥, 𝑦, 𝑧 =
𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be the differentiable vector point function defined over S, then,
S
𝐹. d 𝑠 =
𝑉
𝑑𝑖𝑣 𝐹 𝑑𝑉
Vector Integration
 Stoke’s Theorem :- Let S be an open surface bounded by a closed curve ‘c’ & 𝐹 𝑥, 𝑦, 𝑧 =
𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be a differentiable vector function defined over ‘s’, then 𝑐
𝐹 . 𝑑 𝑟 = 𝑠
𝛻 ×
𝐹. 𝑑 𝑠 = 𝑠
𝛻 × 𝐹 . 𝑛 𝑑𝑠
𝛻 × 𝐹 =
𝑖 𝑗 𝑘
𝜕
𝜕𝑥
𝜕
𝜕𝑦
𝜕
𝜕𝑧
𝐹1 𝐹2 𝐹3
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GATE Engineering Maths : Vector Calculus

  • 1. Section 2 : Calculus Topic 7 : Vector Calculus
  • 2. Vector Basics  Position vector :- The position vector of the point P(x, y, z) in the space is 𝑟 = 𝑥 𝑖 + 𝑦 𝑗 + 𝑧 𝑘 𝑟 = 𝑥2 + 𝑦2 + 𝑧2  In parametric form, 𝑟 = 𝑥 𝑡 𝑖 + 𝑦 𝑡 𝑗 + 𝑧(𝑡) 𝑘  Let, 𝑎 = 𝑎1 𝑖 + 𝑎2 𝑗 + 𝑎3 𝑘, 𝑏 = 𝑏1 𝑖 + 𝑏2 𝑗 + 𝑏3 𝑘 𝑎.𝑏 = 𝑎 𝑏 cos 𝑎. 𝑏 = 𝑎1 𝑏1 + 𝑎2 𝑏2 + 𝑎3 𝑏3 𝑎 × 𝑏 = 𝑎 |𝑏| sin( 𝑎. 𝑏) 𝑛 , where n is vector of unit length perpendicular to the plane containing 𝑎 & 𝑏.  𝑎 × 𝑏 = 𝑖 𝑗 𝑘 𝑎1 𝑎2 𝑎3 𝑏1 𝑏2 𝑏3
  • 3. Vector Basics  Area of ∆𝑂𝐴𝐵 = 1 2 𝑂𝐴 × 𝑂𝐵 = 1 2 𝑎 × 𝑏  Area of ∆𝐴𝐵𝐶 = 1 2 𝐴𝐵 × 𝐴𝐶 = 1 2 (𝑏 − 𝑎) × ( 𝑐 − 𝑎)  Area of parallelogram = | 𝑎 × 𝑏|  Scalar triple product :- 𝑎 × 𝑏 . 𝑐 = 𝑎. 𝑏 × 𝑐 = [ 𝑎 𝑏 𝑐] = 𝑎1 𝑎2 𝑎3 𝑏1 𝑏2 𝑏3 𝑐1 𝑐2 𝑐3  Vector triple product :- 𝑎 × 𝑏 × 𝑐 = 𝑐. 𝑎 𝑏 − 𝑏. 𝑎 𝑐
  • 4. Vector Differentiation  Let 𝑟 𝑡 = 𝑓(𝑡) then, 𝑑 𝑟 𝑑𝑡 = lim ∆𝑡→0 𝑓 𝑡+∆𝑡 − 𝑓(𝑡) ∆𝑡  If t is a time variable then 𝑑 𝑟 𝑑𝑡 represents a velocity vector. 1. 𝑑 𝑟 𝑑𝑡 is a vector in direction of tangent to the curve at that point. 2. If 𝑓(𝑡) is constant in magnitude then 𝐹. 𝑑 𝐹 𝑑𝑡 = 0 3. If 𝑓(𝑡) has constant direction then, 𝐹 × 𝑑 𝐹 𝑑𝑡 = 0
  • 5. Vector Differentiation  Vector differential operator :- 𝛻 (nebla) 𝛻 = 𝑖 𝜕 𝜕𝑥 + 𝑗 𝜕 𝜕𝑦 + 𝑘 𝜕 𝜕𝑧  Gradient of a scalar function :- Let 𝜑(𝑥, 𝑦, 𝑧) be a differentiable scalar point function then gradient of scalar is denoted by grad 𝜑 or 𝛻𝜑 = 𝑖 𝜕𝜑 𝜕𝑥 + 𝑗 𝜕𝜑 𝜕𝑦 + 𝑘 𝜕𝜑 𝜕𝑧  Where, 𝛻𝜑 is vector normal to surface 𝜑.  Unit vector normal to surface 𝜑 can be given as 𝛻𝜑 |𝛻𝜑| .
  • 6. Vector Differentiation  Directional derivative :- The directional derivative of differentiable scalar function 𝜑(𝑥, 𝑦, 𝑧) in the direction of 𝑎 is given by, 𝛻𝜑. 𝑎 |𝑎|  Let 𝑎 = 𝑖, then,  D.D. = 𝛻𝜑. 𝐼 | 𝐼| = ( 𝑖 𝜕𝜑 𝜕𝑥 + 𝑗 𝜕𝜑 𝜕𝑦 + 𝑘 𝜕𝜑 𝜕𝑧 ). 𝑖 = 𝜕𝜑 𝜕𝑥  Angle between surfaces :- It is the angle between the normal to the surfaces at the point of intersection. Let 𝜃 be the angle between the surfaces 𝜑1 𝑥, 𝑦, 𝑧 = 𝐶1 & 𝜑2 𝑥, 𝑦, 𝑧 = 𝐶2 then, cos 𝜃 = 𝛻∅1 𝛻𝜑2 |𝛻∅1| |𝛻𝜑2|
  • 7. Vector Differentiation  Divergence of a vector function :- Let 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be a differential vector point function then, 𝑑𝑖𝑣 𝐹 = 𝛻. 𝐹 = 𝜕𝐹1 𝜕𝑥 + 𝜕𝐹2 𝜕𝑦 + 𝜕𝐹3 𝜕𝑧 Note :- If 𝛻. 𝐹 = 0 then 𝐹 is called solenoidal vector.  Curl of a vector function :- 𝑐𝑢𝑟𝑙 𝐹 = 𝛻 × 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝐹1 𝐹2 𝐹3 Note :- If 𝛻 × 𝐹 = 0 then 𝐹 is called irrotational vector. If 𝑣 = velocity vector and 𝑤 = angular velocity, 𝑤 = 1 2 𝑐𝑢𝑟𝑙 𝑣
  • 8. Vector Differentiation  Scalar Potential Function :- If for every rotational vector, a scalar function 𝜑 exist such that 𝐹 = 𝛻𝜑, then 𝜑 is said to be scalar potential function.  Note :- 1) 𝑐𝑢𝑟𝑙 𝑔𝑟𝑎𝑑 𝜑 = 0 2) 𝑑𝑖𝑣 𝑐𝑢𝑟𝑙 𝐹 = 0 3) 𝑑𝑖𝑣 𝑔𝑟𝑎𝑑 𝜑 = 𝛻 𝛻𝜑 = 𝛻2 𝜑, where 𝛻2= 𝜕2 𝜕𝑥2 + 𝜕2 𝜕𝑦2 + 𝜕2 𝜕𝑧2 (𝛻2 Laplacian Operator)
  • 9. Vector Integration  Line integral :- An integral evaluated over a curve is called line integral.  Let, 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘, be a differentiable point function defined at each point on curve ‘c’ then its line integral is 𝑐 𝐹. 𝑑𝑟 = 𝑐 𝐹1 𝑑𝑥 + 𝐹2 𝑑𝑦 + 𝐹3 𝑑𝑧  If ‘c’ is closed curve  𝑐 𝐹. 𝑑𝑟 Note :- If 𝐹 is irrotational then, the line integral of 𝐹 is independent of path. When, 𝐹 is irrotational  𝑎 𝑏 𝐹. 𝑑𝑟 = 𝜑 𝑏 − 𝜑 𝑎 (Where, 𝜑 is scalar potential function)
  • 10. Vector Integration  Green’s theorem :- Let, M(x, y) & N(x, y) be continuous function having continuous first order partial derivative defined in the closed region R bounded by closed curve ‘c’ then, 𝑐 (𝑀𝑑𝑥 + 𝑁𝑑𝑦) = 𝑅 𝜕𝑁 𝜕𝑥 − 𝜕𝑀 𝜕𝑦 𝑑𝑥 𝑑𝑦  Surface Integral :- Let 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be a differentiable vector point function defined over the surface S then, its surface integration is S 𝐹. d 𝑠 = S 𝐹. 𝑛 ds Where, 𝑛  unit outward drawn normal to the surface
  • 11. Vector Integration  Methods of evaluation of surface integral:- 1. If 𝑅1 is the projection of ‘S’ on to x-y plane then, 𝑠 𝐹 . 𝑛 𝑑𝑠 = 𝑅1 𝐹 . 𝑛 𝑑𝑥 𝑑𝑦 |𝑛 𝑘| 2. If 𝑅2 is the projection of ‘S’ on to y-z plane then, 𝑠 𝐹 . 𝑛 𝑑𝑠 = 𝑅2 𝐹 . 𝑛 𝑑𝑦 𝑑𝑧 |𝑛 𝑖| 3. If 𝑅3 is the projection of ‘S’ on to x-z plane then, 𝑠 𝐹 . 𝑛 𝑑𝑠 = 𝑅3 𝐹 . 𝑛 𝑑𝑧 𝑑𝑥 |𝑛 𝑗|
  • 12. Vector Integration  Let, 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be the differential vector point function defined in volume V, then its volume integral is 𝑉 𝐹 𝑑𝑣  Gauss Divergence Theorem :- Let s be a closed surface enclosing a volume V & 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be the differentiable vector point function defined over S, then, S 𝐹. d 𝑠 = 𝑉 𝑑𝑖𝑣 𝐹 𝑑𝑉
  • 13. Vector Integration  Stoke’s Theorem :- Let S be an open surface bounded by a closed curve ‘c’ & 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be a differentiable vector function defined over ‘s’, then 𝑐 𝐹 . 𝑑 𝑟 = 𝑠 𝛻 × 𝐹. 𝑑 𝑠 = 𝑠 𝛻 × 𝐹 . 𝑛 𝑑𝑠 𝛻 × 𝐹 = 𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝐹1 𝐹2 𝐹3