Calculator Inputs
Example Data Table
| Example vector | Parameter t | x(t) | y(t) | z(t) | dx/dt | dy/dt | dz/dt | Speed |
|---|---|---|---|---|---|---|---|---|
| ⟨3t² + 2sin(t), t³ - cos(t), e^(0.2t)⟩ | 1.5 | 8.74499 | 3.304263 | 1.349859 | 9.141474 | 7.747495 | 0.269972 | 11.985955 |
Formula Used
Let the vector function be r(t) = ⟨x(t), y(t), z(t)⟩. Each component in this calculator uses a mixed model:
f(t) = at³ + bt² + ct + d + psin(ωt) + qcos(ωt) + re^(λt)
First derivative: f′(t) = 3at² + 2bt + c + pωcos(ωt) − qωsin(ωt) + rλe^(λt)
Second derivative: f″(t) = 6at + 2b − pω²sin(ωt) − qω²cos(ωt) + rλ²e^(λt)
Speed: |r′(t)| = √[(dx/dt)² + (dy/dt)² + (dz/dt)²]
Unit tangent: T(t) = r′(t) / |r′(t)|
Curvature: κ(t) = |r′(t) × r″(t)| / |r′(t)|³
2D slope: dy/dx = (dy/dt) / (dx/dt), when dx/dt ≠ 0
How to Use This Calculator
- Select 2D or 3D mode.
- Enter the parameter value where you want the derivative evaluated.
- Set the shared angular frequency ω and exponential rate λ.
- Fill the coefficients for x(t), y(t), and optionally z(t).
- Choose graph start, graph end, and the number of sample points.
- Press Calculate Vector Derivative to show results above the form.
- Review the trajectory plot, derivative plot, tangent vector, speed, and curvature.
- Use the CSV or PDF buttons to export the current results.
Frequently Asked Questions
1. What does vector differentiation measure?
It measures how each component of a vector function changes with respect to a parameter. The derivative often represents velocity, direction change, or rate of motion in geometry, physics, and engineering.
2. Why does the calculator return both first and second derivatives?
The first derivative describes instantaneous change or velocity. The second derivative shows how that change itself varies, which is useful for acceleration, curvature studies, and motion analysis.
3. What is the meaning of the unit tangent vector?
The unit tangent vector points in the direction of motion and has length one. It helps separate direction from speed and is central in curve tracing and kinematics.
4. When is the 2D slope undefined?
The slope dy/dx becomes undefined when dx/dt equals zero at the selected parameter. In that situation, the curve may have a vertical tangent or a stationary horizontal projection.
5. Why are ω and λ shared across components?
Shared frequency and growth rate keep the interface cleaner while still supporting many mixed vector functions. You can still create rich trajectories by assigning different amplitudes in each component.
6. What does curvature tell me?
Curvature measures how sharply the path bends at the chosen parameter value. Larger curvature means tighter turning, while smaller curvature indicates a flatter, straighter local path.
7. Can I model purely polynomial vectors?
Yes. Set the sine, cosine, and exponential coefficients to zero. Then the calculator behaves like a standard polynomial vector differentiation tool with plotted motion and export options.
8. What should I do if my graph looks too rough?
Increase the graph sample points or widen the plotting interval. More sample points produce smoother curves and give a clearer picture of changes across the selected parameter range.