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
- Can be found here: http://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates