Answer
False
Work Step by Step
Step 1: In this problem, we have to determine whether the following statement is true or not: If a vector field is continuous, then its line integral on a smooth curve is a vector. Step 2: Let us start by stating the definition for a line integral: \[ \int_C \mathbf{F} \cdot d\mathbf{r} \] Here \(\mathbf{F}\) is a vector field, and \(d\mathbf{r}\) is an infinitesimal vector that points along the curve \(\mathbf{C}\). Since the dot product between two vectors is always scalar, the line integral is also always scalar. Therefore, the given statement is FALSE. Step 3: Alternatively, if the vector field \(\mathbf{F}\) can be written in the component form as \(\langle M,N \rangle\), where \(M\) and \(N\) are scalar functions, then we can write: \[ \int_C \mathbf{F} \cdot d\mathbf{r} = \int_C \langle M,N \rangle \cdot \langle dx,dy \rangle = \int_C (Md\mathbf{x} + Nd\mathbf{y}) \] The above is a scalar integral; hence, its output is also scalar. Result: FALSE
