Exercise 1.06

In Euclidean three-space, let $p$ be the point with coordinates $(x, y, z) = (1, 0, -1)$. Consider the following curves that pass through $p$: $$ \begin{align} x^{i}(\lambda)&=(\lambda,(\lambda-1)^{2},-\lambda) \\ x^{i}(\mu)&=(\cos\mu,\sin\mu,\mu-1) \\ x^{i}(\sigma)&=(\sigma^{2},\sigma^{3}+\sigma^{2},\sigma) \end{align} $$(a) Calculate the components of the tangent vectors to these curves at p in the coordinate basis $\{\partial_x,\partial_y,\partial_z\}$.
(b) Let $f = x^2 + y^2 - yz$. Calculate $df/d\lambda$, $df/d\mu$ and $df/d\sigma$.

At the point $p=(1,0,-1)$, we have:

$$\lambda= 1\qquad \mu= 0\qquad \sigma= -1$$

a) Tangent vectors at $p$.

$$ \begin{align} \left. \dfrac{dx^{i}(\lambda)}{d\lambda} \right|_{p}&= (1,2(\lambda-1),-1) \Big|_{\lambda=1}=(1,0,-1)  \\ \\ \left. \dfrac{dx^{i}(\mu)}{d\mu} \right|_{p}&= (-\sin\mu,\cos\mu,1) \Big|_{\mu=0}=(0,1,1)  \\ \\ \left. \dfrac{dx^{i}(\sigma)}{d\sigma} \right|_{p}&= (2\sigma,3\sigma^{2}+2\sigma,1) \Big|_{\sigma=-1}=(-2,1,1) \end{align} $$

b) $f$ as a function of $\lambda$, $\mu$ and $\sigma$.

$$ \begin{align} f(\lambda)&= \lambda^{2}+(\lambda-1)^{4}-(\lambda-1)^{2}(-\lambda)=\lambda^4 - 3 \lambda^3 + 5 \lambda^2 - 3 \lambda + 1\\ f(\mu)&=\underbrace{ \cos ^{2}\mu+\sin ^{2}\mu }_{ 1 }-\sin \mu(\mu-1)=1+(1-\mu)\sin \mu \\ f(\sigma)&=\sigma ^{4}+(\sigma^{3}+\sigma^{2})^{2}-(\sigma^{3}+\sigma^{2})\sigma= \sigma^6 + 2 \sigma^5 + \sigma ^{4} - \sigma^{3} \end{align} $$

Taking the derivatives.

$$ \begin{align} \frac{df(\lambda)}{d\lambda}&= 4\lambda^3 - 9 \lambda^2 + 10 \lambda - 3 \\ \frac{df(\mu)}{d\mu}&=(1-\mu)\cos \mu -\sin \mu \\ \frac{df(\sigma)}{d\sigma}&= 6\sigma^5 + 10 \sigma^4 + 4\sigma ^{3} - 3\sigma^{2} \end{align} $$

At the point $p$:

$$ \begin{align} \left. \frac{df(\lambda)}{d\lambda} \right|_{\lambda=1}&= 4 - 9 + 10 - 3 =2\\ \left.\frac{df(\mu)}{d\mu}\right|_{\mu=0}&=\cos 0 -\sin 0 =1\\ \left.\frac{df(\sigma)}{d\sigma}\right|_{\sigma=-1}&= -6 + 10 - 4 - 3=-3 \end{align} $$