Problem understanding vector gradient component output

I'm trying to understand vector analysis and thought I'd generate some plots with matlab and play around with the input to see how it affects the output.

Mathematically I understand the gradient operation used on a scalar vector/matrix will - for each point in the matrix - generate a vector in the direction of the steepest slope.

I did the following

Define a vector / scalar field:

vec3 =  [   2   0   0   0   1       1   1       0   0   0   0;
            0   0   0   1   2       2   2       1   0   0   0;
            0   0   1   2   3       3   3       2   1   0   0;
            0   1   2   3   4       4   4       3   2   1   0;
            1   2   3   4   5.0     5   5.0     4   3   2   1;
            1   2   3   4   5       6   5       4   3   2   1;
            1   2   3   4   5.0     5   5.0     4   3   2   1;      
            0   1   2   3   4       4   4       3   2   1   0;
            0   0   1   2   3       3   3       2   1   0   0;
            0   0   0   1   2       2   2       1   0   0   0;
            0   0   0   0   1       1   1       0   0   0   0]

Then use some commands I found in the livescript examples that I adapted:

X2 = [ -5   -4  -3  -2  -1  0   1   2   3   4   5]
Y2 = X2'
[px, py] = gradient (vec3)
figure
hold on
contour (X2, Y2, vec3,'ShowText','on' )
quiver(X2,Y2,px,py)

contour works just as I expect, it draws isopotential lines just fine.

However, I have a problem understanding the output.

First of all, the lower left corner, would have expected the vectors and isopotential in the upper left corner.

Secondly, when I look at the matlab output

vec3 =

     2     0     0     0     1     1     1     0     0     0     0
     0     0     0     1     2     2     2     1     0     0     0
     0     0     1     2     3     3     3     2     1     0     0
     0     1     2     3     4     4     4     3     2     1     0
     1     2     3     4     5     5     5     4     3     2     1
     1     2     3     4     5     6     5     4     3     2     1
     1     2     3     4     5     5     5     4     3     2     1
     0     1     2     3     4     4     4     3     2     1     0
     0     0     1     2     3     3     3     2     1     0     0
     0     0     0     1     2     2     2     1     0     0     0
     0     0     0     0     1     1     1     0     0     0     0

px =

   -2.0000   -1.0000         0    0.5000    0.5000         0   -0.5000   -0.5000         0         0         0
         0         0    0.5000    1.0000    0.5000         0   -0.5000   -1.0000   -0.5000         0         0
         0    0.5000    1.0000    1.0000    0.5000         0   -0.5000   -1.0000   -1.0000   -0.5000         0
    1.0000    1.0000    1.0000    1.0000    0.5000         0   -0.5000   -1.0000   -1.0000   -1.0000   -1.0000
    1.0000    1.0000    1.0000    1.0000    0.5000         0   -0.5000   -1.0000   -1.0000   -1.0000   -1.0000
    1.0000    1.0000    1.0000    1.0000    1.0000         0   -1.0000   -1.0000   -1.0000   -1.0000   -1.0000
    1.0000    1.0000    1.0000    1.0000    0.5000         0   -0.5000   -1.0000   -1.0000   -1.0000   -1.0000
    1.0000    1.0000    1.0000    1.0000    0.5000         0   -0.5000   -1.0000   -1.0000   -1.0000   -1.0000
         0    0.5000    1.0000    1.0000    0.5000         0   -0.5000   -1.0000   -1.0000   -0.5000         0
         0         0    0.5000    1.0000    0.5000         0   -0.5000   -1.0000   -0.5000         0         0
         0         0         0    0.5000    0.5000         0   -0.5000   -0.5000         0         0         0


py =

   -2.0000         0         0    1.0000    1.0000    1.0000    1.0000    1.0000         0         0         0
   -1.0000         0    0.5000    1.0000    1.0000    1.0000    1.0000    1.0000    0.5000         0         0
         0    0.5000    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000    0.5000         0
    0.5000    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000    1.0000    0.5000
    0.5000    0.5000    0.5000    0.5000    0.5000    1.0000    0.5000    0.5000    0.5000    0.5000    0.5000
         0         0         0         0         0         0         0         0         0         0         0
   -0.5000   -0.5000   -0.5000   -0.5000   -0.5000   -1.0000   -0.5000   -0.5000   -0.5000   -0.5000   -0.5000
   -0.5000   -1.0000   -1.0000   -1.0000   -1.0000   -1.0000   -1.0000   -1.0000   -1.0000   -1.0000   -0.5000
         0   -0.5000   -1.0000   -1.0000   -1.0000   -1.0000   -1.0000   -1.0000   -1.0000   -0.5000         0
         0         0   -0.5000   -1.0000   -1.0000   -1.0000   -1.0000   -1.0000   -0.5000         0         0
         0         0         0   -1.0000   -1.0000   -1.0000   -1.0000   -1.0000         0         0         0

vec3 looks like i entered.

But

Funnily, the -2|-2 gradient is also in the "upper left corner" of px and py.

However, when I look on to the the values of py, i am confused.

I have the feeling they are inversed.

On the quiver plot, the arrows show into the middle of the plot, from all directions (except the corner). Why,

e.g. py for -4,-3 = -0.5

(it is, absolutely seen, 1,3; one column to the right, three rows up, from the lower left)

I would have assumed the y-component of this vector would point upwards, not into the negative direction.

The whole lower half of the py-component is negative, altough vectors are showin up. And the upper half is reversed.

Why is that? I have the feeling i'm missing something. Is there some convention? Or might i have misunderstood some fundamentials here...

Thanks!

Answer 1

[Aditya]

Hi s3bbo,

I understand that you're looking for a clear explanation as to why the output from MATLAB's gradient function may not match the expected orientation in the Cartesian coordinate system and how to correct this.

In MATLAB, matrices are displayed with the first row at the top, which is opposite to the Cartesian coordinate system where the y-axis increases as you move upwards. This difference can lead to confusion when interpreting the output of gradient vectors, as they may appear inverted on the y-axis compared to traditional Cartesian plots.

To address this issue, you can flip the matrix along its vertical axis before performing the gradient operation. This way, the matrix will be oriented correctly with respect to the Cartesian coordinate system, with the first row of the matrix corresponding to the lowest y-value (bottom of the plot) and increasing as you move upwards.

Here's the MATLAB code that flips the matrix “vec3” using the “flipud” function:

% Flip the matrix upside down to align with the Cartesian coordinate system 
vec3_cartesian = flipud(vec3); 

% Now calculate the gradient 
[px, py] = gradient(vec3_cartesian); 

By using “flipud(vec3)”, you create a new matrix “vec3_cartesian” that, when plotted, will have the same orientation as if it were plotted on a standard Cartesian plane. The gradient vectors calculated from vec3_cartesian will then be consistent with the Cartesian coordinate system, thus eliminating the confusion.

For more information on “flipud” function, refer to the MATLAB documentation here:

• https://www.mathworks.com/help/releases/R2019b/matlab/ref/flipud.html

I hope this helps!