<< Calculus`VectorAnalysis`

SetCoordinates[Cartesian[x, y, z]];

Grad[T[x, y]*(E^(-(z^2)))]

Grad[x]

{1, 3, 2}*{a, b, c}

{1, 3, 2} . {a, b, c}

Cross[{1, 3, 2}, {a, b, c}]

{1, 3, 2}×{a, b, c}

xr = r*Cos[ω*t];
yr = r*Sin[ω*t];
vvect = D[{xr, yr, 0}, t]
vel[v_] := (Sqrt[v . v]);

vel[vvect] // Simplify[#, {r > 0, ω > 0}]  &

Simplify[vel[vvect], {r > 0, ω > 0}]

f[x_] := (x^2);
\(\(g := #\^2 &;\)\)
f[2]
g[2]

Pi // N[#, 20] &

pl1 = Plot[Sin[x], {x, 0, 1}]

Show[pl1]

<< Graphics`Arrow`

Graphics[{Arrow[{0, 0}, {1, 1}], Hue[0], 
        Arrow[{ .75,  .25}, { .25,  .75}]}]

Show[
      Graphics[{RGBColor[0, 0, 1], Arrow[{0, 0}, {1, 1}], Hue[0], 
          Arrow[{ .75,  .25}, { .25,  .75}]}]]

krok = 0.01;
pl2 = ListPlot[Table[{x, x}, {x, 0, 1, krok}]];

oabrvenyplot = 
        pl2 /. Point[{x_, y_}] -> {PointSize[0.02], Hue[x], 
              Point[{x, y}]};

    Show[oabrvenyplot]

InputForm[pl2];

Off[General::"spell1"];
er = {r Cos[fi], r Sin[fi]}
ve = {-r ω Sin[fi], 
          r ω Cos[fi]} /. ω -> 1

n = 10;
krok = ((2  Pi)/n);
rozmer = 2;
sipka[r_, fi_] = 
        Graphics[{RGBColor[0, 0, 1], 
            Arrow[er, er + ve, 
              HeadScaling -> Relative]}];
Table[
      Show[sipka[1, fi], 
        PlotRange -> {{-rozmer, rozmer}, {-rozmer, rozmer}}, 
        AspectRatio -> Automatic], {fi, 0, 2  Pi - krok, 
        krok}]

sipky = {sipka[1, 0.1], sipka[1, 0.2]}

Table[
      Show[Table[sipka[pol, fi], {fi, 0, 2  Pi - krok, krok}], 
        PlotRange -> {{-rozmer, rozmer}, {-rozmer, rozmer}}, 
        AspectRatio -> Automatic], {pol, 0.1, 1, 0.1}]

?Circle

Show[
      Table[sipka[pol, fi], {fi, 0, 2  Pi - krok, krok}, {pol, 0.1, 1, 0.1}], 
      Graphics[Circle[{0, 0}, 1]], 
      PlotRange -> {{-rozmer, rozmer}, {-rozmer, rozmer}}, 
      AspectRatio -> Automatic]

pl1 = Plot[x, {x, 0, 1}, AspectRatio -> Automatic]

Graphics[Arrow[{ .75,  .25}, { .25,  .75}]] // FullForm

SetCoordinates[Cylindrical[r, fi, z]];
Evect = {0, 0, Ez[r]};
rotE = Curl[Evect];
Bvect = {0, Bfi[r], 0};
rceveslozkach = 
      Table[rotE[[i]] == -I*ω*Bvect[[i]], {i, 1, 
          3}]
prvniMaxw = rceveslozkach[[2]]

Havect = (1/μ)*Bvect;
rotH = Curl[Havect];
rceveslozkach2 = 
      Table[rotH[[i]] == γ*Evect[[i]], {i, 1, 
          3}]
druhaMaxw = rceveslozkach2[[3]]

naserce = {prvniMaxw, druhaMaxw}

pom=prvniMaxw/.Solve[D[druhaMaxw, r],Ez'[r]][[1]]

res = DSolve[pom, Bfi[r], r] // 
          Simplify[#, {γ > 0, μ > 0, ω > 0}] & // 
        N

pfce[ξ_] := 
        BesselJ[1, (1.  - I)*ξ];
Plot[{Abs[pfce[x]], Re[pfce[x]], Im[pfce[x]]},{x, 0, 7},PlotStyle -> {{GrayLevel[0], Thickness[0.008]}, {Hue[0], 
              Thickness[0.008]}, {Hue[0.63], Thickness[0.008]}},AxesLabel->{"ξ","Abs,Re,Im"},PlotRange -> All]

pfce[ξ_] := 
        BesselY[1, (1.  - I)*ξ];
Plot[{Abs[pfce[x]], Re[pfce[x]], Im[pfce[x]]},{x, 0.01, 5},PlotStyle -> {{GrayLevel[0], Thickness[0.008]}, {Hue[0], 
              Thickness[0.008]}, {Hue[0.63], Thickness[0.008]}},AxesLabel->{"ξ","Abs,Re,Im"},PlotRange -> All]

ClearAll[bfi];
dos = {γ -> 57*(10^6), ω -> 314, μ -> 
            4  Pi*(10^(-7)), r0 -> 3  cm, 
          cm -> 0.01};
bfi[r_] = Bfi[r] /. res[[1]] /. C[2] -> 0 //. 
        dos
resC1 = Solve[bfi[r0] == ((μ*proud)/(2*Pi*r0)), 
            C[1]][[1]] //. dos

prubehB[r_, proud_] = bfi[r] /. resC1

havect = {0, prubehB[r, proud]/μ, 
          0};
jevect = (Curl[havect] // Simplify) //. dos

Plot[{Abs[prubehB[r, 1]], 0}, {r, 0, 3*(10^(-2))}, 
        PlotStyle -> {{Hue[0.63], Thickness[0.008]}, {GrayLevel[0.63], 
              Thickness[0.001]}}, AxesLabel -> {"r", "B[T]"}, 
        PlotRange -> All];

Plot[{Abs[jevect[[3]] /. proud -> 1], 0}, {r, 0, 
          3*(10^(-2))}, 
        PlotStyle -> {{Hue[0.63], Thickness[0.008]}, {GrayLevel[0.63], 
              Thickness[0.001]}}, AxesLabel -> {"r", "Jz[r]"}, 
        PlotRange -> All];

\(\(QV = 
        Abs[jevect[\([3]\)] /. proud -> 1]\^2\/γ /. 
          dos;\)\)
\(\(Plot[{QV, 0}, {r, 0, 3*10\^\(-2\)}, 
        PlotStyle -> {{Hue[0.63], Thickness[0.008]}, {GrayLevel[0.63], 
              Thickness[0.001]}}, 
        AxesLabel -> {"r", \*""QV[W.\!\(m\^\(-3\)\)]""}, 
        PlotRange -> All];\)\)

JacobianMatrix[]
% // MatrixForm
Det[%%] // Simplify

ztraty = delka*2  Pi*NIntegrate[QV*r, {r, 0, 0.03}]

f

vyr[x_] = a*(x^2) + b*x + c;
resabc = Solve[{vyr[x0] == f[x0], 
            vyr[x0 + Δx] == f[x0 + Δx], 
            vyr[x0 - Δx] == 
              f[x0 - Δx]}, {a, b, c}][[1]]

dilciInt = 
      Integrate[
        vyr[x] /. resabc, {x, x0 - Δx, 
          x0 + Δx}]

n = 10;
imax = 3  n;
pom1 = 
        Table[(((-1))^(i + 1)) + 3, {i, 
            imax}];
pom2 = Prepend[Append[pom1, 1], 1];
nx = Length[pom2];
x1 = 0;
x2 = Pi;
poleikxu = 
        x1 + (x2 - x1)*(1. /nx) 
            Range[nx];
mujint[f_] := ((poleikxu[[2]] - poleikxu[[1]])/3)*
          pom2 . (f /@ poleikxu);
mujint[Sin]
\(mujint[#\^3 &]\)
mocnina[x_] := (x^3);
mujint[mocnina]

Integrate[(1/(x^0.5))*Sin[(x^(2 - Cos[x]))], {x, 0, 1}]