[gmd1-A] Vector Analysis I

Clear["Global`*"]

AA={1,2,3}

BB={3,4,5}

vector addition and subtraction

AA+BB

AA-BB

3*AA

dot product

AA.BB

Dot[AA,BB]

BB.AA

Dot[BB,AA]

norm

AA.AA

Norm[AA]

cross product

AA\[Cross]BB

AB=Cross[AA,BB]

BB\[Cross]AA

AA.AB

BB.AB

triple products

CC={3,2,-1}

AA.BB*CC

AA*BB.CC

AA.BB\[Cross]CC

BB.CC\[Cross]AA

CC.AA\[Cross]BB

AA\[Cross](BB\[Cross]CC)

(AA\[Cross]BB)\[Cross]CC

AA.CC*BB-AA.BB*CC

vector functions

xx[t_]={t^4,t*Sin[t],t^2*Exp[-t]}

vv[t_]=D[xx[t],t]

aa[t_]=D[vv[t],t]

Norm[xx[t]]

$Assumptions=t>0;

Norm[xx[t]]//Simplify

FindMaximum[t*Sin[2t]*Exp[-t],{t,1}]

Plot[t*Sin[2t]*Exp[-t],{t,0,3}]

FindMaximum[t*Sin[2t]*Exp[-t],{t,1}]

FindMinimum[t*Sin[2t]*Exp[-t],{t,2}]

FindRoot[t*Sin[2t]*Exp[-t]==0,{t,1.5}]

scalar and vector fields

s[x_,y_,z_]=x*y^2*z^3+Log[x]*Sin[y]*Exp[-z]

v[x_,y_,z_]={x*y*Cos[z],y*z*Sin[x],z*x*Tan[y]}

Grad[s[x,y,z],{x,y,z}]

Div[v[x,y,z],{x,y,z}]

Curl[v[x,y,z],{x,y,z}]

Laplacian[s[x,y,z],{x,y,z}]

Div[Grad[s[x,y,z],{x,y,z}],{x,y,z}]