Divergence Calculator

Divergence Calculator: Explanation and Usage Guide

The Divergence Calculator is an interactive tool designed to calculate the divergence of a three-dimensional vector field. It provides an intuitive way to compute and visualize the divergence of a vector field ( \mathbf{F}(x, y, z) ), offering both the symbolic representation of the divergence and its evaluation at specific points. Additionally, the tool generates a graphical visualization of the vector field to help users gain deeper insights into its behavior.

What is Divergence?

Divergence is a scalar quantity that measures the rate at which a vector field spreads out or converges at a given point. Mathematically, the divergence of a vector field ( \mathbf{F}(x, y, z) = P(x, y, z)\mathbf{i} + Q(x, y, z)\mathbf{j} + R(x, y, z)\mathbf{k} ) is given by:

[ \text{div} \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} ]

• If the divergence is positive, the vector field is spreading out at that point.
• If the divergence is negative, the vector field is converging at that point.
• If the divergence is zero, the field is said to be solenoidal at that point.

This calculator provides both symbolic divergence and an option to evaluate it numerically at specific points.

Features of the Divergence Calculator

• Symbolic Divergence: Automatically computes the partial derivatives of the vector field components and constructs the divergence equation.
• Point Evaluation: Evaluate the divergence numerically at a specific point ( (x, y, z) ).
• Graphical Visualization: Displays a 3D representation of the vector field using Plotly's interactive 3D plotting capabilities.
• Dropdown Examples: Quickly load predefined examples of vector fields for exploration.
• Error Handling: Ensures that invalid or incomplete inputs are handled gracefully.

How to Use the Divergence Calculator

Follow these simple steps to use the calculator effectively:

1. Input the Vector Field:
2. Enter the ( P(x, y, z) ), ( Q(x, y, z) ), and ( R(x, y, z) ) components of the vector field into the respective input boxes.

3. For example:

   • ( P(x, y, z) = \sin(xy) )
   • ( Q(x, y, z) = \cos(xy) )
   • ( R(x, y, z) = e^z )

4. Select an Example:

5. Use the dropdown menu to load predefined examples of vector fields.

6. Specify an Evaluation Point (Optional):

7. If you want to evaluate the divergence at a specific point, enter the values of ( x ), ( y ), and ( z ) in the corresponding fields.

8. Click "Calculate":

9. The calculator will:

   • Compute the symbolic divergence.
   • Evaluate the divergence at the specified point (if provided).
   • Display a step-by-step breakdown of the calculation.
   • Generate a 3D visualization of the vector field.

10. Clear the Inputs:

11. Use the "Clear" button to reset the calculator.

Example Walkthrough

Example Vector Field:

[ \mathbf{F}(x, y, z) = \sin(xy)\mathbf{i} + \cos(xy)\mathbf{j} + e^z\mathbf{k} ]

1. Enter the components:
2. ( P(x, y, z) = \sin(xy) )
3. ( Q(x, y, z) = \cos(xy) )

4. ( R(x, y, z) = e^z )

5. Click "Calculate." The calculator will:

6. Compute the partial derivatives:
   • ( \frac{\partial P}{\partial x} = y\cos(xy) )
   • ( \frac{\partial Q}{\partial y} = -x\sin(xy) )
   • ( \frac{\partial R}{\partial z} = e^z )
7. Combine them to find: [ \text{div} \mathbf{F} = y\cos(xy) - x\sin(xy) + e^z ]

8. If evaluation points ( (x=1, y=1, z=0) ) are provided, the result will be evaluated as: [ \text{div} \mathbf{F}(1, 1, 0) = 1\cdot \cos(1) - 1\cdot \sin(1) + e^0 = \cos(1) - \sin(1) + 1 \approx 1.5403 ]

9. Visualize the 3D vector field generated on the graph.

FAQ

1. What input formats are supported for vector field components?

The calculator supports functions in terms of ( x ), ( y ), and ( z ). Examples include: - Polynomial functions: ( x^2, y^2 + z ) - Trigonometric functions: ( \sin(xy), \cos(z) ) - Exponential functions: ( e^z, x \cdot e^y )

2. What happens if I don’t provide an evaluation point?

If no evaluation point is specified, the calculator will only display the symbolic divergence equation.

3. Can I use this calculator for 2D vector fields?

Yes, simply leave the ( R(x, y, z) ) component empty or set it to zero.

4. How is the 3D vector field visualization generated?

The calculator uses Plotly to create an interactive 3D vector field graph. Each arrow represents the direction and magnitude of the field at a given point.

5. What if my input has an error?

The calculator checks for errors such as missing components or invalid expressions. A descriptive error message will guide you to fix the issue.

Summary

The Divergence Calculator simplifies the process of analyzing vector fields by automating the calculation of divergence and providing a clear visual representation. Whether you're a student, teacher, or professional, this tool is perfect for gaining insights into the behavior of vector fields in 3D space. Start exploring now to unlock the full potential of this powerful calculator!