Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(C_{2}=-C_{1},\) then \(\int_{C_{1}} f(x, y) d s+\int_{C_{2}} f(x, y) d s=0\).

Short answer

The statement is true. When \(C_{2}\) is the reverse orientation of \(C_{1}\), the integrals along these curves are negations of each other, thus their sum is 0.

Step-by-step solution

Understand the concept of Line Integrals

Before diving into the statement, it's crucial to get a firm grasp upon the notion of line integrals. A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. The values of the function are summed to provide a measure of the total influence of the function along the curve.

Comprehend the Reverse Orientation of Curves

When the orientation of a curve \(C_{1}\) described by function \(f(x, y)\) is reversed, meaning you take the curve but trace it in opposite direction, we denote this reversed curve as \(-C_{1}\). In line integrals, reversing the orientation changes the sign of the integral. This property forms the basis of the statement.

Evaluate the statement

Given that \(C_{2}=-C_{1}\), it means \(C_{2}\) is identical to \(C_{1}\) but with reversed orientation. According to our property from Step 2, the line integral along \(C_{2}\) will be exactly the negation of the line integral along \(C_{1}\). Therefore, when you add these two values (\(\int_{C_{1}} f(x, y) d s+\int_{C_{2}} f(x, y) d s\)), the result is indeed 0.