Answer
Result: FALSE
Work Step by Step
Step 1: In this problem, we have to determine whether the following statement is true or not: If \(\mathbf{C}\) is a smooth curve in the $xy$-plane, then the following equation is always true: \[ \int_C \mathbf{F}(\mathbf{x},\mathbf{y}) \cdot d\mathbf{s} = -\int_{-\mathbf{C}} \mathbf{F}(\mathbf{x},\mathbf{y}) \cdot d\mathbf{s} \] Step 2: In order to reduce confusion, let us replace the curve \(-\mathbf{C}\) with \(\mathbf{S}\). Note that: \(-\mathbf{C}\) is simply a curve that has the same shape but the opposite orientation as \(\mathbf{C}\). Now, the equation becomes as follows: \[ \int_C \mathbf{F}(\mathbf{x},\mathbf{y}) \cdot d\mathbf{s} = -\int_{\mathbf{S}} \mathbf{F}(\mathbf{x},\mathbf{y}) \cdot d\mathbf{s} \] Let us consider a function \(\mathbf{F}(\mathbf{x},\mathbf{y})\), that is always positive, since \(\mathbf{d\mathbf{s}}\) is positive by definition, their product, and hence their sum is always positive. Therefore, the given statement is FALSE. Result: FALSE
