potential() compute scalar potential from gradient vector field; Error using sym/potential Second argument must be a vector of variables.

I have a matrix dRho which is a (density) gradient vector field. I want to compute the scalar potential (the density) Rho by means of the matlab-function potential(V,X,Y).

X and Y are my cartesian coordinate vectors:

[m, n] = size(dRho); [1, n] = size(X); [m, 1] = size(Y);

As the function potential() deals with symbolic variables, I convert my quantities to symbolic matrix/vectors with the sym() function, 'f' for floating point precision. (According to numeric to symbolic conversion https://ch.mathworks.com/help/symbolic/numeric-to-symbolic-conversion-1.html#buyp69j-1)

dRho_sym = sym(dRho, 'f');

X_sym = sym(X, 'f');

Y_sym = sym(Y, 'f');

Then I'm trying to compute the potential for each row of my matrix wrt to X_sym, using the respective y value (Y_sym(j)) as base point for the integration, as suggested in the description:

"potential(V,X,Y) computes the potential of vector field V with respect to X using Y as base point for the integration".

for j = 1:m

Rho = potential( dRho_sym(j,:), X_sym, Y_sym(j));

end

I'm getting this error:

Error using sym/potential

Second argument must be a vector of variables.

Are the symbolic numbers not understood as symbolic variables?

Or do I mess up something with the dimensions?

This is a more detailed description in the potential() function:

% P = POTENTIAL(V,X,Y) computes the potential of vector field V with

% respect to X using Y as base point for the integration. If Y is not a

% scalar, then Y must be of the same dimension as V and X. If Y is

% scalar, then Y is expanded into a vector of the same size as X with

% all components equal to Y.

%

% Examples:

% syms x y z; potential([x y z*exp(z)], [x y z])

% returns x^2/2 + y^2/2 + exp(z)*(z - 1).

%

% syms x0 y0 z0; potential([x y z*exp(z)], [x y z], [x0 y0 z0])

% returns x^2/2 - x0^2/2 + y^2/2 - y0^2/2 + exp(z)*(z - 1) -

% exp(z0)*(z0 - 1)

%

% potential([x y z*exp(z)], [x y z], [1,1,1])

% returns x^2/2 + y^2/2 + exp(z)*(z - 1) - 1

%

% potential([x y z*exp(z)], [x y z], 1)

% returns x^2/2 + y^2/2 + exp(z)*(z - 1) - 1

%

% potential([y -x], [x y])

% returns NaN since the potential does not exist.

I will be very gratefull for any help! Also suggestions about how to compute the scalar potential of a gradient vector field by other means are welcome.

Answer 1

[Walter Roberson]

No, the second parameter must be a vector of symbolic names.

When you do not have a starting point, each term in V is integrated with respect to the corresponding variable in the second parameter, and all of the integrals are added.

But you cannot integrate anything with respect to a number, so what you are trying to do cannot work.

[Philipp Bisignano]

x = sym('x', [1 n]);

y = sym('y', [1 m]);

for j = 1:m

Rho_sym(j) = potential(dRho_sym(j,:), x, y(j) );

end

Like this it is compiling. But I'm getting one single value per iteration,

which is function of single entries x1, x2, ..., y1, y2, .. of x and y.

And it takes plenty of computation time.

Could I match the vector x = [x1, x2, ...] and y to my vectors X and Y?

Somehow x1 = X(1), ... ? Would that make any sense to you?