Line integral of a vector field
Accumulate the tangential component of a vector field along a curve.
Line integral of a vector field is one of 1 vector calculus formula in the calculus section of this library, and it is used at university level.
Why line integral of a vector field works
Only the part of the field pointing along the path counts; the sideways part contributes nothing. The dot product with the velocity vector keeps exactly the along-the-path share, and the size of that velocity converts parameter steps into real distance covered.
What each symbol means
$C$ is oriented by parametrization $\mathbf r(t)$.
Line integral of a vector field: when it holds
The field and path should be sufficiently smooth; reversing orientation changes the sign.
When it stops applying
The value depends on the direction of travel and, unless the field is conservative, on the exact route taken. Reversing the path flips the sign, and a second route between the same two endpoints can give a completely different number.
Line integral of a vector field: a worked example
For $\mathbf F=(1,0)$ along $\mathbf r(t)=(t,t)$, $0\le t\le1$, the integral is $1$.
The mistake to avoid
What people do: Integrating the size of the field along the curve instead of the dot product.
Why it goes wrong: That treats a field pushing sideways as if it were helping, and it also loses the cancellation that should happen where the field opposes the motion.
Do this instead: Take the dot product with the velocity vector first, so directions that disagree can cancel as they should.
Line integral of a vector field: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $C$ is oriented by parametrization $\mathbf r(t)$.
- Check the conditions before substituting. The field and path should be sufficiently smooth; reversing orientation changes the sign.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Vector calculus
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where line integral of a vector field comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- The Dot Product — the lesson behind this formula: measure alignment and test orthogonality.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about line integral of a vector field
Does the answer depend on how I parametrize the curve?
No, provided the path is traced once in the same direction. Moving faster shortens the parameter interval and the two effects cancel exactly.
What does this measure physically?
If the field is a force, the integral is the work done travelling the path. If the field is a flow, it measures circulation along that route.
When is the result zero around every closed loop?
When the field is conservative, meaning it is the gradient of some potential function. Then only the two endpoints matter and a loop returns nothing.
How is this different from integrating with respect to arc length?
That version integrates a plain scalar and ignores direction entirely, which is why reversing the curve leaves it unchanged instead of flipping its sign.