Consider a continuous curve $\gamma$ in $\mathbb{R}^3$ parameterized by $\mathbf{r}(t)$ as $t:a \rightarrow b$. Now, it is my understanding that line integrals $$ \int_\gamma \phi \ ds $$ for a scalar field $\phi$ does not depend on the orientation of $\gamma$.
My confusion arises as follows. It is my understanding that $\mathbf{r}(t)$ still corresponds to the curve in question but with reversed orientation if we let $t:b \rightarrow a$. But if this is the case we get $$ \int_\gamma \phi \ ds = \int_a^b \phi \| \dot{\mathbf{r}} \| \ dt \neq \int_b^a \phi \| \dot{\mathbf{r}} \| \ dt = \int_{\gamma_{\text{rev}}} \phi \ ds. $$
Anyone see what I'm missing? Any help is appreciated!
