VECTOR ANALYSIS Dr.
Mohammed Yousuf Kamil
VECTOR DIFFERENTIATION
1. Differentiation of vectors: In Cartesian coordinates, the derivative
of the vector a(u) = axi + ayj + azk is given by
𝑑𝐚 𝑑𝑎𝑥 𝑑𝑎𝑦 𝑑𝑎𝑧
= 𝑖+ 𝑗+ 𝑘
𝑑𝑢 𝑑𝑢 𝑑𝑢 𝑑𝑢
If r(t) = x(t) i+y(t) j+z(t) k, the velocity of the particle is given by the vector
𝑑𝐫 𝑑𝑥 𝑑𝑦 𝑑𝑧
𝐯(𝑡) = = 𝑖+ 𝑗+ 𝑘
𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡
And the acceleration of the particle is given in by
𝑑𝐯 𝑑 2 𝑥 𝑑2𝑦 𝑑2𝑧
𝐚(𝑡) = = 𝑖+ 2𝑗+ 2𝑘
𝑑𝑡 𝑑𝑡 2 𝑑𝑡 𝑑𝑡
Differentiation formulas. If A, B and C are differentiable vector
functions of a scalar u, and Φ is a differentiable scalar function of u, then
𝑑 𝑑𝐀 𝑑𝐁
1. ( 𝐀 + 𝐁) = +
𝑑𝑢 𝑑𝑢 𝑑𝑢
𝑑 𝑑𝐁 𝑑𝐀
2. ( 𝐀 ∙ 𝐁) = 𝐀 ∙ + ∙ 𝐁
𝑑𝑢 𝑑𝑢 𝑑𝑢
𝑑 𝑑𝐁 𝑑𝐀
3. ( 𝐀 × 𝐁) = 𝐀 × + × 𝐁
𝑑𝑢 𝑑𝑢 𝑑𝑢
𝑑 𝑑𝐀 𝑑𝛷
4. (𝛷𝐀) = 𝛷 + 𝐀
𝑑𝑢 𝑑𝑢 𝑑𝑢
𝑑 𝑑𝐂 𝑑𝐁 𝑑𝐀
5. (𝐀 ∙ 𝐁 × 𝐂 ) = 𝐀 ∙ 𝐁 × +𝐀∙ × 𝐂+ ∙𝐁× 𝐂
𝑑𝑢 𝑑𝑢 𝑑𝑢 𝑑𝑢
𝑑 𝑑𝐂 𝑑𝐁 𝑑𝐀
6. (𝐀 × (𝐁 × 𝐂)) = 𝐀 × (𝐁 × ) + 𝐀 × ( × 𝐂) + × (𝐁 × 𝐂)
𝑑𝑢 𝑑𝑢 𝑑𝑢 𝑑𝑢
Partial derivatives of vectors. If A is a vector depending on more than
one scalar variable (x, y, z), then we write A = A(x, y, z). The partial
derivative of A with respect to x, y and z respectively, defined as:
Page 9 Second Class in Department of Physics
VECTOR ANALYSIS Dr. Mohammed Yousuf Kamil
𝜕𝐀 𝐀(𝑥 + ∆𝑥, 𝑦, 𝑧) − 𝐀(𝑥, 𝑦, 𝑧)
= lim
𝜕𝑥 ∆𝑥→0 ∆𝑥
𝜕𝐀 𝐀(𝑥, 𝑦 + ∆𝑦, 𝑧) − 𝐀(𝑥, 𝑦, 𝑧)
= lim
𝜕𝑦 ∆𝑦→0 ∆𝑦
𝜕𝐀 𝐀(𝑥, 𝑦, 𝑧 + ∆𝑧) − 𝐀(𝑥, 𝑦, 𝑧)
= lim
𝜕𝑧 ∆𝑧→0 ∆𝑧
Higher derivatives can be defined as
𝜕2𝐀 𝜕 𝜕𝐀 𝜕2𝐀 𝜕 𝜕𝐀 𝜕2𝐀 𝜕 𝜕𝐀
= ( ), = ( ), = ( )
𝜕𝑥 2 𝜕𝑥 𝜕𝑥 𝜕𝑦 2 𝜕𝑦 𝜕𝑦 𝜕𝑧 2 𝜕𝑧 𝜕𝑧
𝜕2𝐀 𝜕 𝜕𝐀 𝜕2𝐀 𝜕 𝜕𝐀 𝜕2𝐀 𝜕2𝐀
= ( ), = ( ), =
𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥
Thus if A and B are functions of x,y,z then,
𝜕 𝜕𝐁 𝜕𝐀
1. (𝐀 ∙ 𝐁) = 𝐀 ∙ + ∙ 𝐁
𝜕𝑥 𝜕𝑥 𝜕𝑥
𝜕 𝜕𝐁 𝜕𝐀
2. (𝐀 × 𝐁) = 𝐀 × + × 𝐁
𝜕𝑥 𝜕𝑥 𝜕𝑥
𝜕2 𝜕 𝜕 𝜕 𝜕𝐁 𝜕𝐀
3. (𝐀 ∙ 𝐁) = { (𝐀 ∙ 𝐁)} = {𝐀 ∙ + ∙ 𝐁}
𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑥
𝜕2𝐁 𝜕𝐀 𝜕𝐁 𝜕𝐀 𝜕𝐁 𝜕2𝐀
=𝐀 ∙ + ∙ + ∙ + ∙𝐁
𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥
Differentials of vectors follow rules similar to those of elementary
calculus.
1. If A = Ali + A2j + A3k , then dA = dA1i + dA2j + dA3k
2. d(A . B) = A . dB + dA . B
3. d(A × B) = A × dB + dA × B
4. If A = A(x,y,z), then
𝜕𝐀 𝜕𝐀 𝜕𝐀
𝑑𝐀 = 𝑑𝑥 + 𝑑𝑦 + 𝑑𝑧 , ⟹ (𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙)
𝜕𝑥 𝜕𝑦 𝜕𝑧
Page 10 Second Class in Department of Physics
VECTOR ANALYSIS Dr. Mohammed Yousuf Kamil
𝑑𝐀 𝜕𝐀 𝜕𝐀 𝑑𝑦 𝜕𝐀 𝑑𝑧
= +( ) +( ) , ⟹ (𝑡𝑜𝑡𝑎𝑙 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑓𝑜𝑟 𝑥 )
𝑑𝑥 𝜕𝑥 𝜕𝑦 𝑑𝑥 𝜕𝑧 𝑑𝑥
2. Vector operators: Central to all these differential operations is the
vector operator 𝛁, and in Cartesian coordinates is defined by
𝜕 𝜕 𝜕
𝛁=𝑖 +𝑗 +𝑘
𝜕𝑥 𝜕𝑦 𝜕𝑧
2.1. The Gradient: The gradient of a scalar field φ(x, y, z) is defined by
𝜕∅ 𝜕∅ 𝜕∅
𝑔𝑟𝑎𝑑 ∅ = 𝛁∅ = 𝑖 +𝑗 +𝑘
𝜕𝑥 𝜕𝑦 𝜕𝑧
Note that 𝛁∅ defines a vector field.
Any scalar field ∅ for which 𝛁∅ = 0 is said to be constant.
2.2.The Divergence: The divergence of a vector field 𝑨(𝑥, 𝑦, 𝑧) = 𝐴𝑥 𝑖 +
𝐴𝑦 𝑗 + 𝐴𝑧 𝑘 is defined by:
𝜕𝐴𝑥 𝜕𝐴𝑦 𝜕𝐴𝑧
𝑑𝑖𝑣 𝐀 = 𝛁 ∙ 𝐀 = + +
𝜕𝑥 𝜕𝑦 𝜕𝑧
Note that 𝛁 ∙ 𝐀 defines a scalar field. Also that 𝛁 ∙ 𝐀 ≠ 𝐀 ∙ 𝛁
Any vector field A for which 𝛁 ∙ 𝐀 = 0 is said to be solenoidal.
2.3.The Curl: The curl of a vector field 𝑨(𝑥, 𝑦, 𝑧) is defined by
𝜕𝐴𝑧 𝜕𝐴𝑦 𝜕𝐴𝑥 𝜕𝐴𝑧 𝜕𝐴𝑦 𝜕𝐴𝑥
𝑐𝑢𝑟𝑙 𝐀 = 𝛁 × 𝐀 = ( − )𝑖 + ( − )𝑗 + ( − )𝑘
𝜕𝑦 𝜕𝑧 𝜕𝑧 𝜕𝑥 𝜕𝑥 𝜕𝑦
𝑖 𝑗 𝑘
𝜕 𝜕 𝜕
𝐴𝑙𝑠𝑜, 𝛁 × 𝐀 = || ||
𝜕𝑥 𝜕𝑦 𝜕𝑧
𝐴𝑥 𝐴𝑦 𝐴𝑧
Page 11 Second Class in Department of Physics
VECTOR ANALYSIS Dr. Mohammed Yousuf Kamil
Note that 𝛁 × 𝐀 defines a vector field. Any vector field A for which
𝛁 × 𝐀 = 0 is said to be irrotational.
2.4.The Laplacian: The Laplacian of a scalar field φ(x, y, z) is defined by
2
𝜕2 𝜕2 𝜕2
𝛁 =𝛁∙𝛁= 2+ 2+ 2
𝜕𝑥 𝜕𝑦 𝜕𝑧
Note that 𝛻 2 defines a scalar field.
Formulas involving 𝛁. If A and B are differentiable vector functions, and
𝜙 and 𝜓 are differentiable scalar functions of position (x, y, z), then
1. 𝛁(𝜙 + 𝜓) = 𝛁𝜙 + 𝛁ψ
2. 𝛁 ∙ (𝐀 + 𝐁) = 𝛁 ∙ 𝐀 + 𝛁 ∙ 𝐁
3. 𝛁 × (𝐀 + 𝐁) = 𝛁 × 𝐀 + 𝛁 × 𝐁
4. 𝛁 ∙ (ϕ𝐀) = (𝛁ϕ) ∙ 𝐀 + ϕ(𝛁 ∙ 𝐀)
5. 𝛁(𝜙𝜓) = 𝜙 𝛁𝜓 + 𝜓 𝛁𝜙
6. 𝛁 × (ϕ𝐀) = (𝛁ϕ) × 𝐀 + ϕ(𝛁 × 𝐀)
7. 𝛁 ∙ (𝐀 × 𝐁) = 𝐁 ∙ (𝛁 × 𝐀) − 𝐀 ∙ (𝛁 × 𝐁)
8. 𝛁 × (𝐀 × 𝐁) = (𝐁 ∙ 𝛁)𝐀 − 𝐁(𝛁 ∙ 𝐀) − (𝐀 ∙ 𝛁)𝐁 + 𝐀(𝛁 ∙ 𝐁)
9. 𝛁(𝐀 ∙ 𝐁) = (𝐁 ∙ 𝛁)𝐀 + (𝐀 ∙ 𝛁)𝐁 + 𝐁 × (𝛁 × 𝐀) + 𝐀 × (𝛁 × 𝐁)
10. 𝛁 × (𝛁ϕ) = 0 , 𝑐𝑢𝑟𝑙 𝑔𝑟𝑎𝑑 𝜙 = 0
11. 𝛁 ∙ (𝛁 × 𝐀) = 0 , 𝑑𝑖𝑣 𝑐𝑢𝑟𝑙 𝐀 = 0
12. 𝛁 × (𝛁 × 𝐀) = 𝛁(𝛁 ∙ 𝐀) − 𝛁 2 𝐀
13. 𝛁 ∙ (𝛁𝜙 × 𝛁𝜓) = 0
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