Calculating the gradient of a surface

[kakarot1905]

Hi

z(x,y,t)=a sin(ωt) sin(k/Lx*pi*x) sin(l/Ly*pi*y)

a = Amplitude
ω = Frequency
k and l are constants
Lx = Length in x direction
Ly = Length in y direction


How can I find [using an equation] the slope of the surface [ie the gradient] at any given point on the surface?

I know how to do it in the x direction and y direction independently:
dz/dx for x direction and dz/dy for y direction
But how do I combine these two things?

Thanks in Advance

 

[tiny-tim]

hi kakarot1905!

the downhill https://www.physicsforums.com/library.php?do=view_item&itemid=11" (∂z/∂x,∂z/∂y) is what you need

 

[kakarot1905]

hi kakarot1905!

the downhill https://www.physicsforums.com/library.php?do=view_item&itemid=11" (∂z/∂x,∂z/∂y) is what you need

Thanks tiny-tim

I can calculate:

(dz/dx) = (k\pia)/Lx sin(\omegat) cos (k\pix)/Lx
and
(dz/dy) = (l\pib)/Ly sin(\omegat) cos (l\piy)/Ly

But what is dz/dt?


This is why I need to calculate the slope of the surface (z):

I am trying to calculate the weight [W] of a particle on the dynamic surface (z)
Because of the slope'ness' of the surface z, the acceleration of particle (parallel to the surface) is affected
So in order to calculate the surface parallel Weight, I need to W*(Gradient of the slope)

The above link by tiny-tim, helps me calculate the gradient in vector form, but how do convert it into a value to multiply it with W? Do I take the abs?


Please Help, Thanks

 

[tiny-tim]

hi kakarot1905!

This is why I need to calculate the slope of the surface (z):

I am trying to calculate the weight [W] of a particle on the dynamic surface (z)

i really don't understand what you're trying to do

i don't see the relevance of the component W_||, but it would be W times the sin of tan^-1 of the gradient

 

[kakarot1905]

hi kakarot1905!


i really don't understand what you're trying to do

i don't see the relevance of the component W_||, but it would be W times the sin of tan^-1 of the gradient

Thanks for the suggestion tiny tim.

I got my plotting software [mathematica] to automatically calculate the gradient of the z function so I don't have to worry about taking dz/dx...

This is the code i used: [mathematica]

(gradf[x_, y_, t_] = {D[z[x, y, t], x], D[z[x, y, t], y], 
   D[z[x, y, t], t]})