Green's Theorem Calculator
This Green's Theorem calculator uses Green's Theorem to calculate either the line integral of a vector field along a closed curve.
Result
Enter values and click "Compute" to see the result.
Steps to Use the Green's Theorem Calculator
- Enter the components of the vector field M(x, y) and N(x, y) in the input fields.
- Specify the parametric equations x(t) and y(t) that define the boundary curve.
- Provide the range for the parameter t, ensuring the curve is closed if you want to apply Green's Theorem.
- Click "Compute" to calculate the line integral and verify if Green's Theorem holds.
Additional Features
- Visualization: The calculator provides a graphical representation of the curve and the vector field for better understanding.
- Step-by-Step Solution: Displays detailed calculations, including partial derivatives and intermediate steps.
- Error Checking: Warns if the curve is not closed or if inputs are inconsistent.
Common Vector Fields
| Vector Field |
Applications |
| → M(x, y) = (x, y) |
Radial fields |
| → M(x, y) = (-y, x) |
Rotational fields |
| → M(x, y) = (x2, y2) |
Quadratic scaling fields |
| → M(x, y) = (x * y, x2) |
Mixed fields |
| → M(x, y) = (-y2, x3) |
Higher-order fields |
Green's Theorem Statement
Green's Theorem states:
∫C → F • d→ r = ∫R (∂N/∂x - ∂M/∂y) dA
Where:
- C: The closed curve bounding the region R.
- → F = (M(x, y), N(x, y)): The vector field.
- d→ r: The differential vector along the curve C.
- ∂N/∂x, ∂M/∂y: The partial derivatives of the components of the vector field.
Important Notes
- Ensure that the curve C is closed to apply Green's Theorem. If it is not closed, the result will only represent the line integral.
- For simple closed curves, Green's Theorem can be used to easily compute the circulation or flux of a vector field.
- The theorem is valid only for regions R where the vector field is continuously differentiable.
- Orientation matters: Ensure the curve is traversed in the positive (counter-clockwise) direction when applying Green's Theorem.
FAQs About Green's Theorem Calculator
- What is Green's Theorem used for?
Green's Theorem is commonly used in physics and engineering to compute circulation, flux, or work along a closed path.
- Can the calculator handle non-closed curves?
Yes, but the results will only represent the line integral, and Green's Theorem will not apply.
- Does the calculator support custom vector fields?
Yes, you can input any vector field in terms of x and y components.
- What happens if I input a non-differentiable vector field?
The calculator may provide a warning or fail to compute the integral, as Green's Theorem requires the vector field to be continuously differentiable.
- How can I verify my results?
The calculator shows intermediate steps and a comparison between the line integral and the double integral to validate Green's Theorem.
Example Problem
Problem: Compute the circulation of the vector field F(x, y) = (-y, x) along the unit circle x(t) = cos(t), y(t) = sin(t), 0 ≤ t ≤ 2π, using Green's Theorem.
Solution:
- Step 1: Input M(x, y) = -y and N(x, y) = x into the calculator.
- Step 2: Specify x(t) = cos(t) and y(t) = sin(t).
- Step 3: Provide the range t: [0, 2π].
- Step 4: Compute: The line integral equals 2π, confirming the result using Green's Theorem.
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