General Vector Line Integral Calculator
This formula defines the line integral of a vector field along a general parameterized curve C, representing quantities like work done by a force.
Formula first
Overview
The integral evaluates the accumulation of a vector field along a path by taking the dot product of the field with the tangent vector of the curve. By parameterizing the curve as r(t), the problem is reduced to a standard definite integral with respect to the parameter t. This method is fundamental for calculating flux, circulation, and work in conservative or non-conservative fields.
Symbols
Variables
F = Vector Field, r(t) = Parameterization
Apply it well
When To Use
When to use: Use this formula when you need to calculate the work done by a force field along a specific path or the circulation of a fluid flow along a curve.
Why it matters: It serves as the foundation for physical concepts such as energy transfer, electric potential, and fluid dynamics, connecting local vector fields to global path-dependent results.
Avoid these traps
Common Mistakes
- Forgetting to multiply by the derivative of the parameterization (r'(t)) inside the integral.
- Failing to substitute the parameterized variables into the vector field F, leaving x, y, and z as independent variables.
One free problem
Practice Problem
Calculate the work done by the force field F = <y, x> along the curve r(t) = <cos(t), sin(t)> for t from 0 to pi.
Solve for:
Hint: Compute r'(t) = <-sin(t), cos(t)> and dot it with F(r(t)) = <sin(t), cos(t)>.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning.