How to plot grad(f(x,y))?

Question

I want to calculate and plot a gradient of any scalar function of two variables. If you really want a concrete example, lets say f=x^2+y^2 where x goes from -10 to 10 and same for y. How do I calculate and plot grad(f)? The solution should be vector and I should see vector lines. I am new to python so please use simple words.

EDIT:

@Andras Deak: thank you for your post, i tried what you suggested and instead of your test function (fun=3*x^2-5*y^2) I used function that i defined as V(x,y); this is how the code looks like but it reports an error

import numpy as np
import math 
import sympy
import matplotlib.pyplot as plt

def V(x,y):
    t=[]
    for k in range (1,3): 
        for l in range (1,3):
            t.append(0.000001*np.sin(2*math.pi*k*0.5)/((4*(math.pi)**2)* (k**2+l**2)))  
            term = t* np.sin(2 * math.pi * k * x/0.004) * np.cos(2 * math.pi * l * y/0.004)
            return term 
    return term.sum()

x,y=sympy.symbols('x y')
fun=V(x,y)
gradfun=[sympy.diff(fun,var) for var in (x,y)]
numgradfun=sympy.lambdify([x,y],gradfun)

X,Y=np.meshgrid(np.arange(-10,11),np.arange(-10,11))
graddat=numgradfun(X,Y)
plt.figure()
plt.quiver(X,Y,graddat[0],graddat[1])
plt.show()

AttributeError: 'Mul' object has no attribute 'sin'

And lets say I remove sin, I get another error:

TypeError: can't multiply sequence by non-int of type 'Mul'

I read tutorial for sympy and it says "The real power of a symbolic computation system such as SymPy is the ability to do all sorts of computations symbolically". I get this, I just dont get why I cannot multiply x and y symbols with float numbers.

What is the way around this? :( Help please!

UPDATE

@Andras Deak: I wanted to make things shorter so I removed many constants from the original formulas for V(x,y) and Cn*Dm. As you pointed out, that caused the sin function to always return 0 (i just noticed). Apologies for that. I will update the post later today when i read your comment in details. Big thanks!

UPDATE 2 I changed coefficients in my expression for voltage and this is the result:

It looks good except that the arrows point in the opposite direction (they are supposed to go out of the reddish dot and into the blue one). Do you know how I could change that? And if possible, could you please tell me the way to increase the size of the arrows? I tried what was suggested in another topic (Computing and drawing vector fields):

skip = (slice(None, None, 3), slice(None, None, 3))

This plots only every third arrow and matplotlib does the autoscale but it doesnt work for me (nothing happens when i add this, for any number that i enter) You were already of huge help , i cannot thank you enough!

Answer 1

Here's a solution using sympy and numpy. This is the first time I use sympy, so others will/could probably come up with much better and more elegant solutions.

import sympy

#define symbolic vars, function
x,y=sympy.symbols('x y')
fun=3*x**2-5*y**2

#take the gradient symbolically
gradfun=[sympy.diff(fun,var) for var in (x,y)]

#turn into a bivariate lambda for numpy
numgradfun=sympy.lambdify([x,y],gradfun)

now you can use numgradfun(1,3) to compute the gradient at (x,y)==(1,3). This function can then be used for plotting, which you said you can do.

For plotting, you can use, for instance, matplotlib's quiver, like so:

import numpy as np
import matplotlib.pyplot as plt

X,Y=np.meshgrid(np.arange(-10,11),np.arange(-10,11))
graddat=numgradfun(X,Y)

plt.figure()
plt.quiver(X,Y,graddat[0],graddat[1])
plt.show()

UPDATE

You added a specification for your function to be computed. It contains the product of terms depending on x and y, which seems to break my above solution. I managed to come up with a new one to suit your needs. However, your function seems to make little sense. From your edited question:

t.append(0.000001*np.sin(2*math.pi*k*0.5)/((4*(math.pi)**2)* (k**2+l**2)))
term = t* np.sin(2 * math.pi * k * x/0.004) * np.cos(2 * math.pi * l * y/0.004)

On the other hand, from your corresponding comment to this answer:

V(x,y) = Sum over n and m of [Cn * Dm * sin(2pinx) * cos(2pimy)]; sum goes from -10 to 10; Cn and Dm are coefficients, and i calculated that CkDl = sin(2pik)/(k^2 +l^2) (i used here k and l as one of the indices from the sum over n and m).

I have several problems with this: both sin(2*pi*k) and sin(2*pi*k/2) (the two competing versions in the prefactor are always zero for integer k, giving you a constant zero V at every (x,y). Furthermore, in your code you have magical frequency factors in the trigonometric functions, which are missing from the comment. If you multiply your x by 4e-3, you drastically change the spatial dependence of your function (by changing the wavelength by roughly a factor of a thousand). So you should really decide what your function is.

So here's a solution, where I assumed

V(x,y)=sum_{k,l = 1 to 10} C_{k,l} * sin(2*pi*k*x)*cos(2*pi*l*y), with
C_{k,l}=sin(2*pi*k/4)/((4*pi^2)*(k^2+l^2))*1e-6

This is a combination of your various versions of the function, with the modification of sin(2*pi*k/4) in the prefactor in order to have a non-zero function. I expect you to be able to fix the numerical factors to your actual needs, after you figure out the proper mathematical model.

So here's the full code:

import sympy as sp
import numpy as np
import matplotlib.pyplot as plt

def CD(k,l):
    #return sp.sin(2*sp.pi*k/2)/((4*sp.pi**2)*(k**2+l**2))*1e-6
    return sp.sin(2*sp.pi*k/4)/((4*sp.pi**2)*(k**2+l**2))*1e-6

def Vkl(x,y,k,l):
    return CD(k,l)*sp.sin(2*sp.pi*k*x)*sp.cos(2*sp.pi*l*y)

def V(x,y,kmax,lmax):
    k,l=sp.symbols('k l',integers=True)
    return sp.summation(Vkl(x,y,k,l),(k,1,kmax),(l,1,lmax))


#define symbolic vars, function
kmax=10
lmax=10
x,y=sp.symbols('x y')
fun=V(x,y,kmax,lmax)

#take the gradient symbolically
gradfun=[sp.diff(fun,var) for var in (x,y)]

#turn into bivariate lambda for numpy
numgradfun=sp.lambdify([x,y],gradfun,'numpy')
numfun=sp.lambdify([x,y],fun,'numpy')

#plot
X,Y=np.meshgrid(np.linspace(-10,10,51),np.linspace(-10,10,51))
graddat=numgradfun(X,Y)
fundat=numfun(X,Y)

hf=plt.figure()
hc=plt.contourf(X,Y,fundat,np.linspace(fundat.min(),fundat.max(),25))
plt.quiver(X,Y,graddat[0],graddat[1])
plt.colorbar(hc)
plt.show()

I defined your V(x,y) function using some auxiliary functions for transparence. I left the summation cut-offs as literal parameters, kmax and lmax: in your code these were 3, in your comment they were said to be 10, and anyway they should be infinity.

The gradient is taken the same way as before, but when converting to a numpy function using lambdify you have to set an additional string parameter, 'numpy'. This will alow the resulting numpy lambda to accept array input (essentially it will use np.sin instead of math.sin and the same for cos).

I also changed the definition of the grid from array to np.linspace: this is usually more convenient. Since your function is almost constant at integer grid points, I created a denser mesh for plotting (51 points while keeping your original limits of (-10,10) fixed).

For clarity I included a few more plots: a contourf to show the value of the function (contour lines should always be orthogonal to the gradient vectors), and a colorbar to indicate the value of the function. Here's the result:

The composition is obviously not the best, but I didn't want to stray too much from your specifications. The arrows in this figure are actually hardly visible, but as you can see (and also evident from the definition of V) your function is periodic, so if you plot the same thing with smaller limits and less grid points, you'll see more features and larger arrows.