📝 Path independence of line integrals (14 MCQs)

📖 From Calculus • 16. Topics in vector Calculus • 14 questions available

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What is Path independence of line integrals?

Path independence of line integrals:
A line integral is path independent if the vector field is conservative, i.e., $\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$ for any closed curve.

Example:
For $\mathbf{F} = \langle y, x \rangle$ , $\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$ for all closed curves, confirming path independence.

Reason:
This property ensures energy conservation and allows defining scalar potentials, simplifying many physics and engineering problems.

Easy

Medium

Hard

📝 All Path independence of line integrals MCQs

Q1. A vector field $F$ has the property that the line integral from $A$ to $B$ gives the same value along every smooth path in a simply connected region. Which conclusion is most justified?

A.The field must have constant magnitude

B.The field is conservative on the region ✅

C.Every path between

A

and

B

must have the same length

D.The field must be perpendicular to every path

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Path independence means that the line integral depends only on the initial and terminal points, not on the route taken. In a simply connected region, this behavior is equivalent to the existence of a potential function whose gradient equals the vector field, so the field is conservative.

Q2. Suppose $F$ is conservative and $C_1$ and $C_2$ are two piecewise smooth curves joining the same points $P$ and $Q$ . If $\int_{C_1}F\cdot dr=7$ , what is $\int_{C_2}F\cdot dr$ ?

A.It must be

-7

B.It must be

0

C.It must be

7

✅

D.It depends on the lengths of

C_1

and

C_2

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: For a conservative vector field, the line integral is independent of the path and depends only on the endpoints. Since both curves start at $P$ and end at $Q$ , their integrals must be identical. Therefore, the second path also produces an integral of $7$ , regardless of its shape or length.

Q3. Let $F(x,y)=(2xy+y^2, x^2+2xy)$ . A student wants to determine whether the integral from $(0,0)$ to $(1,1)$ is path independent. What is the most efficient first test in the plane?

A.Compare the lengths of several possible paths

B.Check whether the mixed partial derivatives of the components agree on the relevant region ✅

C.Evaluate the integral along one randomly selected path

D.Check whether

F

has constant magnitude

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For a sufficiently smooth planar field on an appropriate simply connected region, equality of the cross-partial derivatives provides a powerful test for conservativeness. Here, $\partial P/\partial y=2x+2y$ and $\partial Q/\partial x=2x+2y$ , so the field passes this local test.

Q4. Consider $F=(y,x)$ on the entire plane. One student computes the integral along the straight segment from $(0,0)$ to $(1,1)$ , while another uses a broken path through $(1,0)$ . What should happen?

A.The two values must agree because the field is conservative ✅

B.The two values must differ because the paths have different lengths

C.The two values agree only if both paths are straight

D.The result cannot be predicted without evaluating both paths

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The field $F=(y,x)$ is conservative because it is the gradient of $xy$ . Its potential function is $f(x,y)=xy$ , so any path from $(0,0)$ to $(1,1)$ gives $f(1,1)-f(0,0)=1$ . Thus the different geometric routes produce the same integral.

Q5. A field is known to be conservative in a region, and a particle moves from $A$ to $B$ along a highly curved trajectory instead of a straight line. Which modelling interpretation is correct?

A.The work changes because the particle travels farther

B.The work is determined only by the initial and final positions ✅

C.The work is always zero for a curved trajectory

D.The work depends only on the particle's average speed

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: When the force field is conservative, the work done between two fixed positions depends only on those positions. The trajectory can be curved, long, or irregular without changing the work. This is a major modelling advantage because the detailed motion need not be known to determine the work.

Q6. Suppose a force field is conservative and a particle travels from $A$ to $B$ , then returns from $B$ to $A$ along a different route. What is the total work done by the field?

A.It must be positive

B.It must be negative

C.It must be zero ✅

D.It depends on the lengths of the two routes

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: For a conservative field, the work from $A$ to $B$ depends only on the endpoints. The return trip from $B$ to $A$ has exactly the opposite value. Therefore, even though the particle may use completely different routes in the two directions, the total work around the resulting closed path is zero.

Q7. A student calculates a line integral along one convenient path and concludes that the result is valid for every path because the field components look similar in form. What is the main logical flaw?

A.A single path can establish path independence only if conservativeness has already been justified ✅

B.A convenient path can never be used for a line integral

C.Only circular paths can establish path independence

D.Path independence depends only on the endpoint coordinates being positive

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Evaluating one path does not prove that all paths give the same result. To replace an arbitrary path by a convenient one, the student must first establish path independence, typically by proving that the field is conservative under the necessary domain conditions. Without that justification, the conclusion is unsupported.

Q8. A graph shows several directed curves connecting $P$ to $Q$ . Curves $C_1$ and $C_2$ stay inside a region where a vector field is conservative, while $C_3$ leaves that region. Which comparison is guaranteed?

A.All three integrals must be equal

B.Only the integrals along

C_1

and

C_2

are guaranteed to be equal ✅

C.Only

C_2

and

C_3

are guaranteed to be equal

D.None of the integrals can be compared

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Path independence is guaranteed only within a domain where the field is known to be conservative. Since $C_1$ and $C_2$ remain inside that valid region and share the same endpoints, their integrals agree. The behavior of $C_3$ cannot automatically be included because it leaves the region where the property was established.

Q9. A contour map represents a scalar potential $f$ , and three curves connect the same points $P$ and $Q$ . Curve $C_1$ crosses many contour lines, $C_2$ follows a nearly constant contour before reaching $Q$ , and $C_3$ takes a direct route. Which statement about $\int_C\nabla f\cdot dr$ is correct?

C_1

always gives the largest value because it crosses more contours

C_2

gives zero because it follows a contour for part of the motion

C_3

gives the largest value because it is shortest

D.All three give the same value determined by the change in

f

from

P

Q

✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: The gradient field $\nabla f$ is conservative, so its line integral is independent of path. The integral equals the potential difference $f(Q)-f(P)$ . Crossing more contour lines does not inherently increase the integral, and following a contour temporarily only contributes zero during that portion rather than making the entire integral zero.

Q10. A field is defined on a region containing a hole. Its curl is zero everywhere in the region, yet a student concludes that every line integral between two points is path independent. Which issue must be examined before accepting the conclusion?

A.Whether the field has zero magnitude

B.Whether the region is simply connected ✅

C.Whether the endpoints have equal coordinates

D.Whether every path is a straight line

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A zero-curl condition alone does not always guarantee global path independence when the domain has holes. The topology of the region matters. In a simply connected domain, suitable smoothness together with zero curl can establish conservativeness, but a punctured or otherwise non-simply-connected region can allow nonzero circulation around a hole.

Q11. Two methods are proposed for finding the work from $P$ to $Q$ . Method I evaluates a complicated curved path directly. Method II finds a potential function $f$ and computes $f(Q)-f(P)$ . When is Method II mathematically justified and preferable?

A.Whenever the path is curved

B.Whenever the field is conservative on a domain containing the relevant paths ✅

C.Only when

P

and

Q

lie on the same horizontal line

D.Only when the field has constant components

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The potential-function method is justified when the vector field is conservative on the relevant domain. In that case, the line integral is determined solely by endpoint values, making $f(Q)-f(P)$ much more efficient than parameterizing a complicated path and integrating directly.

Q12. A model for energy transfer uses $F(x,y)=(3x^2+2y, 2x+4y)$ . An engineer wants to know whether the work from one location to another depends on the route. What conclusion follows from comparing the cross-partials?

A.The field is conservative because both cross-partials equal

2

✅

B.The field is not conservative because both components contain

x

C.The field is conservative only along horizontal paths

D.The field is conservative only when

x=y

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Let $P=3x^2+2y$ and $Q=2x+4y$ . Then $\partial P/\partial y=2$ and $\partial Q/\partial x=2$ . Since the field is smooth on the entire plane, the matching cross-partials establish conservativeness there. Consequently, work between two fixed locations is independent of the route.

Q13. A student claims that if a line integral around every small closed loop is zero, then the integral between any two points must be path independent, regardless of the shape of the domain. Which evaluation is strongest?

A.The claim is always true because small loops determine all larger loops

B.The claim can fail globally if the domain contains a hole that prevents certain loops from being contracted ✅

C.The claim is false because closed-loop integrals are unrelated to path independence

D.The claim is true only for circular domains

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Local zero circulation does not automatically imply global path independence on arbitrary domains. A region with a hole can contain loops that cannot be continuously shrunk to a point. A field may have zero curl locally while still producing nonzero circulation around such a hole, preventing global path independence.

Q14. A researcher has measured that the work from $A$ to $B$ is $12$ units along several very different routes. The routes all lie inside a connected region, but no mathematical test of the field has been performed. Which conclusion is safest?

A.The field is definitely conservative everywhere

B.The measured equality provides evidence of path independence between

A

and

B

, but does not by itself prove global conservativeness ✅

C.The field must have zero curl everywhere

D.The work must be

12

for every pair of points in the region

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Repeated agreement along several paths supports the hypothesis that the integral may be path independent between the tested endpoints. However, finite experimental evidence does not prove a global mathematical property. Conservativeness requires stronger justification, such as a valid potential function or appropriate derivative and domain analysis.

🔗 Related Topics (MCQs)

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