Answer
True
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 level curve for \(f\), then the following line integral is \(0\): \[ \int_C \nabla f \cdot d\mathbf{r} \] Step 2: Let us start by making the following observations: (1) A level curve for a scalar function is always normal to its gradient vector function. (2) \(d\mathbf{r}\) is an infinitesimal vector that points along the curve \(\mathbf{C}\). (3) The dot product between two normal vectors is \(0\). Therefore, by using (1), (2), and (3), we can conclude that the given statement is TRUE. Result: \[ TRUE \]
