GATE Engineering Maths : Vector Calculus
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1. This document covers key concepts in vector calculus including vector basics, vector differentiation, and vector integration. It defines concepts like position vectors, gradients, divergence, curl, line integrals, and surface integrals. 2. Formulas are provided for calculating directional derivatives, divergence, curl, line integrals, surface integrals, and theorems like Green's theorem and Gauss's divergence theorem. 3. Vector operations like dot products, cross products, and triple products are defined along with their geometric interpretations and formulas for calculation.
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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