PARTIAL DERIVATIVES USING   		MATHEMATICA

\(Find\ partial\ derivatives

      to\ second\ order\ for\ \ f \((x, y)\) = x^2 y^3 Sin[xy]

      \((the\ easiest\ way\ to\ obtain\ the\ partial\ derivative\ 
operator\ 
          is\ to\ pull\ it\ from\ the\ basic\ input\ 
\(tableau--\) if\ this
          \ is\ not\ currently\ open, \ 

        you\ can\ open\ it\ by\ clicking\ on\ BasicInput\ 
under\ the\ File\ 
          menu, \ Palettes\ submenu)\)\)

Define f

f[x, y] := x^2  y^3  Sin[x*y]

Compute first partial derivatives with respect to x and y
           (bring down the (∂_\[Placeholder]) \[Placeholder]  operator from the BasicInput 
tableau and then click on the lower square to enter the 
variable of differentiation (x or y) and click on 
the second square to enter the function
to be differentiated)

\(g[x, y] := ∂\_x\ f[x, y]; 

    h[x, y] := ∂\_y\ f[x, y]; 
g[x, y]\)h[x, y]

Compute second partial derivatives

(∂_(x, x))f[x, y]

(∂_(y, y)) f[x, y]

(∂_(x, y))f[x, y]

(∂_(y, x))f[x, y]

Alternatively these may be computed as first partials of g 
        and h

(∂_x) g[x, y]

(∂_y) h[x, y]

(∂_x) h[x, y]

(∂_y) g[x, y]

Equations involving partial derivatives

Laplace' s equationVerify that u (x, y) = Sin[x] Cosh[y] + Cos[x] Sinh[y]is a solution of Laplace' s equation

u[x, y] := Sin[x] Cosh[y] + Cos[x] Sinh[y]

(∂_(x, x))u[x, y] + (∂_(y, y))u[x, y]

Plot the graph of u

Plot3D[Evaluate[u[x, y]], {x, -2*Pi, 2*Pi}, {y, -3, 3}]

Wave equationVerify that v (t, x) = Sin[x - 2 t] + Log[x + 2 t] is a solution of the wave equation for t and x positive

v[t, x] := Sin[x - 2  t] + Log[x + 2  t]

(∂_(t, t))v - (∂_(x, x))v

Plot the graph of v

Plot3D[Evaluate[v[t, x]], {t, 0.1, 5}, {x, 0, 10}]

Heat Equation \(Verify\ that\ w \((t, x, y, z)\) = 

          Exp[\(-\((x^2 + y^2 + z^2)\)\)/2 t]/Sqrt[2 t] is\ a\ solution\) of the heat equation for t > 0

w[t, x, y, z] := Exp[-(x^2 + y^2 + z^2)/2  t]/Sqrt[2*Pi*t]

(∂_t) w - (∂_(x, x))w - (∂_(y, y))w - 
      (∂_(z, z))w

\(Plot\ the\ graph\ of\ w\ as\ a\ function\ of\ r = 

        Sqrt[x^2 + y^2 + z^2] . \ What\ do\ you\ notice\ about\ the\)crosssections of this surface for t or r fixed?

p[t, r] := Exp[-r^2/2  t]/Sqrt[2*Pi*t]\(Plot3D[Evaluate[p[t, r]], {t, 0.1, 2}, {r, 0, 7}, 
      ViewPoint -> {1.5, 4, 1}]
\)