I need help or a thorough proof on understanding the following theoretical lemma regarding integral curves and fields. This is a part of the "Introduction to Partial Differential Equations with Applications - Zachmanoglou".
First of all, where to begin from for proving part (a) of the lemma-protasis ?
Secondly, I can't comprehend part (b).
Let $C$ be an integral curve of the vector field $V = (P,Q,R)$ and suppose that $C$ is given parametrically by the equations $$x=x(t), \; \; y = y(t), \; \; z = z(t) \; ; \; t \in I$$ where the functions $x(t), y(t)$ and $z(t)$ are in $C^1(I)$ and the tangent vector $$T(t) = \bigg(\frac{\mathrm{d}x(t)}{dt},\frac{\mathrm{d}y(t)}{dt}, \frac{\mathrm{d}z(t)}{dt}\bigg)$$ never vanishes for $t \in I$.
(a) Show that there exists a function $μ(t)$ in $C^1(I)$ such that for every $t \in I$ it is : $$μ(t) \neq 0 \; \; \text{and} \; \; V(x(t),y(t),z(t)) = μ(t)T(t)$$
(b) Let $t = τ(t)$ be a solution of the differential equation $$\frac{\mathrm{d}t}{\mathrm{d}τ}=μ(t)$$ where $τ$ varies over some interval $I'$ as $t$ varies over $I$. Set $$\bar{x}(τ)=x(t(τ)), \; \bar{y}(τ)=y(t(τ)), \; \bar{z}(τ)=\bar{z}(t(τ))$$ Show that in terms of the new parametric representation $$x = \bar{x}(τ), \; \; y = \bar{y}(τ), \; \; z = \bar{z}(τ) \; ; \; t \in I'$$ the curve $C$ is a solution of the system of equations associated with $V$ $$\frac{\mathrm{d}x}{\mathrm{d}τ}=P, \; \frac{\mathrm{d}y}{\mathrm{d}τ}=Q, \; \frac{\mathrm{d}z}{\mathrm{d}τ}=R$$