Vector Analysis


4 vector operations

Addition of two vectors

  • Commutative: $\textbf{A + B = B + A}$
  • Associative: $\textbf{(A + B) + C = A + (B + C)}$

Multiplication by a scalar

  • Distributive: $a(\textbf{A + B}) = a\textbf{A} + a\textbf{B}$, where a is a scalar

Dot product of two vectors (Also known as scalar product)

  • $\textbf{A . B} = \left|A\right|\left|B\right|cos \theta$
  • Commutative: $\textbf{A . B = B . A}$
  • Distributive: $\textbf{A . (B + C) = A.B + A.C}$
  • Geometrically, A . B is product of A times projection of B along A.
  • For any vector A, $\textbf{A . A }= \left|\textbf{A}\right|^{2}$
  • If A and B are perpendicular, then $\textbf{A.B = 0}$

Cross product of two vectors (Also known as vector product)

  • $\textbf{A} \times \textbf{B} = \left|\textbf{A}\right|\left|\textbf{B}\right|sin \theta \hat{\textbf{n}}$, where $\hat{\textbf{n}}$ is the unit vector pointing perpendicularly to the plane of A and B.
  • Distributive: $\textbf{A} \times \textbf{(B + C) = (A} \times \textbf{B) + (A} \times \textbf{C)}$
  • Not commutative: $(\textbf{B} \times \textbf{A}) = – (\textbf{A} \times \textbf{B})$
  • Geometrically, $ \left|\textbf{A} \times \textbf{B}\right|$ is area of parallelogram generated by A and B.
  • If A and B are parallel, then $\textbf{A} \times \textbf{B} = \textbf{0}$

 

Triple product

Scalar triple product

  • $\textbf{A} . (\textbf{B} \times \textbf{C})$
  • Geometrically, $\left|\textbf{A} . (\textbf{B} \times \textbf{C})\right|$ is the volume of the parallelepiped generated by A, B and C, since $ \left|\textbf{B} \times \textbf{C}\right|$ is area of base, and $ \left|\textbf{A}cos \theta\right|$ is the altitude.
  • $\textbf{A} . (\textbf{B} \times \textbf{C}) = \textbf{B} . (\textbf{C} \times \textbf{A}) = \textbf{C} . (\textbf{A} \times \textbf{B})$

Vector triple product

  • $\textbf{A} \times (\textbf{B} \times \textbf{C})$
  • BAC-CAB rule: $\textbf{A} \times (\textbf{B} \times \textbf{C}) = \textbf{B} (\textbf{A} . \textbf{C}) – \textbf{C} (\textbf{A} . \textbf{B})$
  • Note that: $(\textbf{A} \times \textbf{B}) \times \textbf{C} = – \textbf{C} \times (\textbf{A} \times \textbf{B}) = – \textbf{A} (\textbf{B} . \textbf{C}) + \textbf{B} (\textbf{A} . \textbf{C})$

Operator $\nabla$

$\nabla = \hat{x} \frac{\partial}{\partial x} + \hat{y} \frac{\partial}{\partial y} + \hat{z} \frac{\partial}{\partial z}$

Gradient Of T

$\nabla T = \frac{\partial T}{\partial x} \hat{x} + \frac{\partial T}{\partial y} \hat{y} + \frac{\partial T}{\partial z} \hat{z} $

  • The gradient $\nabla T$ points in direction of maximum increase of the function T.
  • The magnitude $\left| \nabla T \right|$ gives the slope (rate of increase) along this maximal direction.

Divergence

$\nabla . v = \frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z}$

Curl

$\nabla \times F =$   

$\nabla \times F = (\frac{\partial F_{z}}{\partial y} – \frac{\partial F_{y}}{\partial z}) \textbf{i} + (\frac{\partial F_{x}}{\partial z} – \frac{\partial F_{z}}{\partial x}) \textbf{j} + (\frac{\partial F_{y}}{\partial x} – \frac{\partial F_{x}}{\partial y}) \textbf{j}$

Rules:

Sum rule:

  • $\nabla (f + g) = \nabla f + \nabla g$
  • $\nabla . (A + B) = (\nabla . A) + (\nabla . B)$
  • $\nabla \times (A + B) = (\nabla \times A) + (\nabla \times B)$

Rule for multiplying by a constant:

  • $\nabla (kf) = k \nabla f$
  • $\nabla . (kA) = k(\nabla . A)$
  • $\nabla \times (kA) = k (\nabla \times A)$

Product Rule:

Two for gradients:

  • $\nabla (fg) = f \nabla g + g \nabla f$
  • $\nabla (A . B) = A \times (\nabla \times B) + B \times (\nabla \times A) + (A . \nabla) B + (B . \nabla) A$

Two for divergence:

  • $\nabla . (fA) = f (\nabla . A) + A . (\nabla f)$
  • $\nabla (A \times B) = B . (\nabla \times A) – A . (\nabla \times B)$

Two for curl:

  • $\nabla \times (fA) = f (\nabla \times A) – A \times (\nabla f)$
  • $\nabla \times (A \times B) = (B . \nabla) A – (A . \nabla) B + A (\nabla . B) – B (\nabla . A)$

 

Quotient Rule

  • $\nabla (\frac{f}{g}) = \frac{ g\nabla f – f \nabla g}{g^{2}}$
  • $\nabla . (\frac{A}{g}) = \frac{g(\nabla . A) – A . (\nabla g)}{g^{2}}$
  • $\nabla \times (\frac{A}{g}) = \frac{g (\nabla \times A) + A \times (\nabla g)}{g^{2}}$

 

Second Derivatives

Laplacian

  • $\nabla ^{2} T = \nabla . (\nabla T) = \frac{\partial ^{2}T}{\partial x^{2}} + \frac{\partial ^{2} T}{\partial y^{2}} + \frac{\partial ^{2}T}{\partial z^{2}}$

Curl of a gradient is always zero

  • $\nabla \times (\nabla T) = 0$

Divergence of a curl is always zero

  • $\nabla . (\nabla \times v) = 0$

Curl-of-curl

  • $\nabla \times (\nabla \times v) = \nabla (\nabla . v) – \nabla ^{2} v$

Vector derivatives in cylindrical and spherical coordinates

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