To Represent the Gradient of Scalar Field and Divergence and Curl of Vector Field using Scilab

1. Aim

To visualize and understand the concepts of gradient of scalar fields, Divergence, and curl of vector fields using Scilab computational tools.

2. Apparatus Used

• Computer with Scilab installed (Version 6.0 or later recommended)
• Scilab documentation for plotting and vector field operations
• Mathematical equations and test cases prepared for demonstration
• Projection device (optional, for demonstration purposes)

3. Diagram

Below are the sample diagrams representing vector fields and scalar fields:

4. Theory

Vector calculus is a branch of mathematics that studies differentiation and integration of vector fields. It provides the mathematical framework for modeling various physical phenomena such as fluid dynamics, electromagnetism, and thermodynamics.

Scalar Field

A scalar field assigns a scalar value to each point in space. Examples include temperature distribution, pressure field, and electric potential.

Vector Field

A vector field assigns a vector to each point in space. Examples include velocity field in fluid flow, electric field, and magnetic field.

Gradient of a Scalar Field

The gradient of a scalar field is a vector field whose magnitude represents the maximum rate of change of the scalar field, and its direction is the direction of maximum increase.

Divergence of a Vector Field

The divergence of a vector field measures the net flow of the field through a small region around each point. A positive divergence indicates a source, while a negative divergence indicates a sink.

Curl of a Vector Field

The curl of a vector field measures the field's tendency to rotate around a point. It is a vector quantity whose direction represents the axis of rotation according to the right-hand rule.

Physical Interpretation

• Gradient: Points in the direction of steepest ascent of a scalar field
• Divergence: Represents sources or sinks in a vector field (like fluid flow or electric field)
• Curl: Represents rotational behavior in a vector field (like fluid vortices or magnetic fields)

5. Formulas

6. Procedure

1. Setup:
   • Launch Scilab on your computer
   • Create a new script file (.sce extension)
2. Define the Domain:
   • Create a mesh grid for the domain of interest (usually 2D or 3D)
   • Example code for creating a 2D mesh grid:

     // Create a mesh grid
     x = linspace(-5, 5, 50);
     y = linspace(-5, 5, 50);
     [X, Y] = meshgrid(x, y);

3. Define a Scalar Field:
   • Choose a scalar function to analyze (e.g., $f(x,y) = x^2 + y^2$)
   • Example code:

     // Define a scalar field
     Z = X.^2 + Y.^2; // Example scalar field: f(x,y) = x² + y²

4. Calculate the Gradient:
   • Compute the partial derivatives to find the gradient
   • Example code:

     // Calculate gradient components using numerical differentiation
     [dZdx, dZdy] = gradient(Z, x, y);

5. Visualize the Scalar Field and its Gradient:
   • Plot the scalar field as a surface plot
   • Overlay gradient vectors on a contour plot
   • Example code:

     // Plot scalar field as a surface
     figure(1);
     surf(X, Y, Z);
     title('Scalar Field f(x,y) = x² + y²');
     xlabel('x'); ylabel('y'); zlabel('f(x,y)');
     
     // Plot gradient vectors on contour plot
     figure(2);
     contour(X, Y, Z, 20);
     hold on;
     xq = X(1:3:$, 1:3:$);
     yq = Y(1:3:$, 1:3:$);
     u = dZdx(1:3:$, 1:3:$);
     v = dZdy(1:3:$, 1:3:$);
     champ(xq, yq, u, v);
     title('Gradient of Scalar Field ∇f');
     xlabel('x'); ylabel('y');

6. Define a Vector Field:
   • Choose a vector field to analyze (e.g., $\vec{F}(x,y) = (y, -x)$)
   • Example code:

     // Define a vector field
     Fx = Y; // Example vector field: F(x,y) = (y, -x)
     Fy = -X;

7. Calculate the Divergence:
   • Compute the partial derivatives and sum them to find the divergence
   • Example code:

     // Calculate divergence
     [dFxdx, temp] = gradient(Fx, x, y);
     [temp, dFydy] = gradient(Fy, x, y);
     div = dFxdx + dFydy;

8. Calculate the Curl (for 2D vector field):
   • Compute the z-component of the curl for a 2D field
   • Example code:

     // Calculate curl (z-component for 2D fields)
     [dFxdy, temp] = gradient(Fx, y, x);
     [temp, dFydx] = gradient(Fy, y, x);
     curl_z = dFydx - dFxdy;

9. Visualize the Vector Field, Divergence, and Curl:
   • Plot the vector field using arrows
   • Visualize divergence and curl using color maps
   • Example code:

     // Plot vector field
     figure(3);
     xq = X(1:3:$, 1:3:$);
     yq = Y(1:3:$, 1:3:$);
     u = Fx(1:3:$, 1:3:$);
     v = Fy(1:3:$, 1:3:$);
     champ(xq, yq, u, v);
     title('Vector Field F(x,y) = (y, -x)');
     xlabel('x'); ylabel('y');
     
     // Plot divergence
     figure(4);
     surf(X, Y, div);
     title('Divergence of Vector Field (∇·F)');
     xlabel('x'); ylabel('y'); zlabel('∇·F');
     
     // Plot curl (z-component)
     figure(5);
     surf(X, Y, curl_z);
     title('Curl of Vector Field (z-component) (∇×F)_z');
     xlabel('x'); ylabel('y'); zlabel('(∇×F)_z');

10. Analysis:
    • Interpret the results by analyzing the plots
    • Verify the calculated values with theoretical expectations
11. Save Results:
    • Export the plots as image files
    • Save the script file for future reference

7. Observation Table

8. Calculations

Here are calculations for some example fields:

Example 1: Scalar Field $f(x,y) = x^2 + y^2$

Example 2: Vector Field $\vec{F}(x,y) = (y, -x)$

Example 3: Vector Field $\vec{F}(x,y,z) = (x^2, y^2, z^2)$

9. Results

From the experiments performed, we observed the following:

1. The gradient of a scalar field points in the direction of steepest ascent. For example, for $f(x,y) = x^2 + y^2$, the gradient vectors point radially outward from the origin, which matches our theoretical understanding.
2. The divergence of vector field $\vec{F}(x,y) = (y, -x)$ is zero everywhere, indicating that this field is incompressible (no sources or sinks).
3. The curl of vector field $\vec{F}(x,y) = (y, -x)$ is constant and equal to $-2\hat{k}$, indicating uniform clockwise rotation throughout the field.
4. The visualization of vector field $\vec{F}(x,y) = (x, y)$ shows outward flow from the origin, consistent with its positive divergence value of 2.
5. For the vector field $\vec{F}(x,y,z) = (x^2, y^2, z^2)$, the divergence increases as we move away from the origin, and the curl is zero everywhere, confirming it is an irrotational (conservative) field.

These observations verify the fundamental principles of vector calculus and demonstrate the utility of Scilab for visualizing and calculating field properties.

10. Precautions

1. Ensure that the domain is properly defined with appropriate resolution for accurate visualization.
2. For complex scalar and vector fields, increase the mesh density for more accurate derivative computations.
3. Be careful with the scaling of vectors when visualizing gradient and vector fields.
4. When plotting multiple fields on the same figure, use appropriate scales and legends for clarity.
5. Numerical differentiation may introduce errors; compare numerical results with analytical solutions when possible.
6. For 3D fields, consider using slices or 2D projections to visualize the field properties effectively.
7. Save workspace and script files periodically to prevent data loss.
8. Verify the sign conventions used in the curl and gradient calculations.
9. Check that the units are consistent throughout all calculations.
10. For large datasets, optimize the code to improve performance.

11. Viva Voice Questions