Math Notes

David Wallace Croft

2005-02-18

Definitions

1. Taylor Series
   $x\left(t+\Delta t\right)=\underset{k=0}{\overset{\infty }{\Sigma }}\frac{{\Delta t}^k}{k!}{x}^{\left(k\right)}\left(t\right)$
2. Exponential Function
   $e^{\Delta t}=\underset{k=0}{\overset{\infty }{\Sigma }}\frac{{\Delta t}^k}{k!}$
3. Matrix Multiplication
   ${\mathrm{<html:b>A</html:b>}}_{\mathrm{m}xn}*{\mathrm{<html:b>B</html:b>}}_{\mathrm{n}xp}={[\underset{k=1}{\overset{n}{\Sigma }}{a}_{ik}*b_{kj}]}_{mxp}$
4. Gradient

   $\nabla f\left(\mathrm{<html:b>x</html:b>}\right)=\frac{\partial {f\left(\mathrm{<html:b>x</html:b>}\right)}_{1x1}}{{\partial \mathrm{<html:b>x</html:b>}}_{\mathrm{n}x1}}={\left[\begin{array}{cccc}\frac{\partial f\left(\mathrm{<html:b>x</html:b>}\right)}{{\partial x}_1}& \frac{\partial f\left(\mathrm{<html:b>x</html:b>}\right)}{{\partial x}_2}& ...& \frac{\partial f\left(\mathrm{<html:b>x</html:b>}\right)}{{\partial x}_{\mathrm{n}}}\end{array}\right]}_{1x\mathrm{n}}$

5. Jacobian Matrix

   $\frac{\partial {\mathrm{<html:b>f</html:b>}\left(\mathrm{<html:b>x</html:b>}\right)}_{\mathrm{m}x1}}{{\partial \mathrm{<html:b>x</html:b>}}_{\mathrm{n}x1}}={\left[\begin{array}{cccc}\frac{\partial f_1\left(\mathrm{<html:b>x</html:b>}\right)}{{\partial x}_1}& \frac{\partial f_1\left(\mathrm{<html:b>x</html:b>}\right)}{{\partial x}_2}& ...& \frac{\partial f_1\left(\mathrm{<html:b>x</html:b>}\right)}{{\partial x}_{\mathrm{n}}}\\ \frac{\partial f_2\left(\mathrm{<html:b>x</html:b>}\right)}{{\partial x}_1}& \frac{\partial f_2\left(\mathrm{<html:b>x</html:b>}\right)}{{\partial x}_2}& ...& \frac{\partial f_2\left(\mathrm{<html:b>x</html:b>}\right)}{{\partial x}_{\mathrm{n}}}\\ ...& ...& ...& ...\\ \frac{\partial f_{\mathrm{m}}\left(\mathrm{<html:b>x</html:b>}\right)}{{\partial x}_1}& \frac{\partial f_{\mathrm{m}}\left(\mathrm{<html:b>x</html:b>}\right)}{{\partial x}_2}& ...& \frac{\partial f_{\mathrm{m}}\left(\mathrm{<html:b>x</html:b>}\right)}{{\partial x}_{\mathrm{n}}}\end{array}\right]}_{\mathrm{m}x\mathrm{n}}$

6. Hessian Matrix

Identities

7. Vector Norm Squared to Dot Product ${\Vert \mathrm{<html:b>x</html:b>}\Vert }^2={\mathrm{<html:b>x</html:b>}}^T*\mathrm{<html:b>x</html:b>}$

8. Vector Derivative

   $\frac{\partial ({\mathrm{<html:b>x</html:b>}}^T*\mathrm{<html:b>x</html:b>})}{\partial \mathrm{<html:b>x</html:b>}}=2*{\mathrm{<html:b>x</html:b>}}^T$

Links

• Character Entity References in HTML 4
• Exponentional Function
• MathML Basics
• MathML Fundamentals