# 数学代写|数学分析代写Mathematical Analysis代考|The Cross Product in R

## 数学代写|数学分析代写Mathematical Analysis代考|The Cross Product in R

The notion of cross product of two vectors in $\mathbb{R}^3$, which we introduce here, will be used in particular in the next two sections.

Let $u=\left(u_1, u_2, u_3\right), v=\left(v_1, v_2, v_3\right)$ be vectors in $\mathbb{R}^3$. The cross product of $u$ and $v$, written $u \wedge v$, is the vector of $\mathbb{R}^3$
$$u \wedge v=\left|\begin{array}{ll} u_2 & v_2 \ u_3 & v_3 \end{array}\right| e_1+\left|\begin{array}{ll} u_3 & v_3 \ u_1 & v_1 \end{array}\right| e_2+\left|\begin{array}{ll} u_1 & v_1 \ u_2 & v_2 \end{array}\right| e_3,$$
where $e_1, e_2, e_3$ are the elements of the canonical basis of $\mathbb{R}^3\left(e_1=(1,0,0)\right.$ etc. $)$. The components of $u \wedge v$ are the $2 \times 2$ minors of the matrix with columns $u$ and $v$
$$\left(\begin{array}{ll} u_1 & v_1 \ u_2 & v_2 \ u_3 & v_3 \end{array}\right)$$
taken with alternating signs,,+-+ . Operationally, (6.33) is computable as the formal expansion, along the first column, of the $3 \times 3$ determinant
$$\left|\begin{array}{lll} e_1 & u_1 & v_1 \ e_2 & u_2 & v_2 \ e_3 & u_3 & v_3 \end{array}\right|=e_1\left|\begin{array}{ll} u_2 & v_2 \ u_3 & v_3 \end{array}\right|-e_2\left|\begin{array}{ll} u_1 & v_1 \ u_3 & v_3 \end{array}\right|+e_3\left|\begin{array}{ll} u_1 & v_1 \ u_2 & v_2 \end{array}\right| .$$
Below we list the cross product’s main properties. Let $\left(u_1, u_2, u_3\right),\left(v_1, v_2, v_3\right)$ and $\left(w_1, w_2, w_3\right)$ be the components of $u, v, w \in \mathbb{R}^3$.
Proposition 1 For any $u, v, w \in \mathbb{R}^3$ and $\lambda, \mu \in \mathbb{R}$,
\begin{aligned} & u \wedge v=-v \wedge u \ & (\lambda u+\mu w) \wedge v=\lambda(u \wedge v)+\mu(w \wedge v) ; \ & (u \wedge v, w)=\left|\begin{array}{lll} u_1 & v_1 & w_1 \ u_2 & v_2 & w_2 \ u_3 & v_3 & w_3 \end{array}\right| ; \ & (u \wedge v, u)=0, \quad(u \wedge v, v)=0 . \ & \end{aligned}
$$u \wedge v=0 \quad \text { if and only if } u \text { and } v \text { are linearly dependent; (6.37) }$$

## 数学代写|数学分析代写Mathematical Analysis代考|Biregular Curves in R3: Curvature

In this section, and in the next, $\gamma:[a, b] \rightarrow \mathbb{R}^3$ will indicate a regular curve in $\mathbb{R}^3$ parametrised by arclength $s$. We want to introduce two characteristic quantities of a curve, the curvature and the torsion. They not only describe the curve’s geometric features, irrespective of coordinate changes or reparametrisations. As we will see, they completely determine the curve itself up to rigid motions.

First of all we remind the differentiation rules for the inner product and the cross product in $\mathbb{R}^3$. Suppose $u(t), v(t): I \rightarrow \mathbb{R}^3$ are differentiable functions on the interval $I$, of respective components $\left(u_i(t)\right),\left(v_i(t)\right)$. For the inner product we showed in $(6.24)$ that
$$\frac{d}{d t}(u(t), v(t))=\left(u^{\prime}(t), v(t)\right)+\left(u(t), v^{\prime}(t)\right)$$
We recall in particular that if $w(t): I \rightarrow \mathbb{R}^3$ is differentiable on the interval $I \subseteq \mathbb{R}$ and $|w(t)|=1$ for any $t \in I$, the vector $w(t)$ is at any $t \in I$ orthogonal to the vector $w^{\prime}(t)$ :
$$\left(w^{\prime}(t), w(t)\right)=0, \quad \forall t \in I$$
The derivative of the cross product equals
$$\frac{d}{d t} u(t) \wedge v(t)=u^{\prime}(t) \wedge v(t)+u(t) \wedge v^{\prime}(t)$$
Differentiating for instance the first component of $u(t) \wedge v(t)$ (see the definition of the cross product in the previous section) gives
\begin{aligned} \frac{d}{d t}\left|\begin{array}{ll} u_2(t) & v_2(t) \ u_3(t) & v_3(t) \end{array}\right| & =\frac{d}{d t}\left(u_2(t) v_3(t)-u_3(t) v_2(t)\right)= \ & =\left(u_2^{\prime}(t) v_3(t)-u_3^{\prime}(t) v_2(t)\right)+\left(u_2(t) v_3^{\prime}(t)-u_3(t) v_2^{\prime}(t)\right), \end{aligned}
which is exactly the first component of $u^{\prime}(t) \wedge v(t)+u(t) \wedge v^{\prime}(t)$. Similarly, the other components of $d(u \wedge v) / d t$ and $u^{\prime} \wedge v+u \wedge v^{\prime}$ coincide.

Returning to the curve $\gamma:[a, b] \rightarrow \mathbb{R}^3$, in this section we shall assume that the function $\gamma(s)$ is $C^2$. We shall take a regular curve $\gamma: I=[a, b] \rightarrow \mathbb{R}^3$ of class $C^2(I)$ parametrised by arclength $s$.

# 数学分析代考

## 数学代写|数学分析代写Mathematical Analysis代考|The Cross Product in R

$$u \wedge v=\left|u_2 \quad v_2 u_3 \quad v_3\right| e_1+\left|u_3 \quad v_3 u_1 \quad v_1\right| e_2+\left|u_1 \quad v_1 u_2 \quad v_2\right| e_3,$$

$$\left(\begin{array}{llllll} u_1 & v_1 & u_2 & v_2 & u_3 & v_3 \end{array}\right)$$

$\left|e_1 \quad u_1 \quad v_1 e_2 \quad u_2 \quad v_2 e_3 \quad u_3 \quad v_3\right|=e_1\left|u_2 \quad v_2 u_3 \quad v_3\right|-e_2\left|u_1 \quad v_1 u_3 \quad v_3\right|$

$u \wedge v=-v \wedge u \quad(\lambda u+\mu w) \wedge v=\lambda(u \wedge v)+\mu(w \wedge v) ;(u \wedge v, w)=\mid u_1 \quad v_1$
$u \wedge v=0 \quad$ if and only if $u$ and $v$ are linearly dependent; (6.37)

## 数学代写|数学分析代写Mathematical Analysis代考|Biregular Curves in R3: Curvature

$$u \wedge v=\left|u_2 \quad v_2 u_3 \quad v_3\right| e_1+\left|u_3 \quad v_3 u_1 \quad v_1\right| e_2+\left|u_1 \quad v_1 u_2 \quad v_2\right| e_3,$$

$$\left(\begin{array}{llllll} u_1 & v_1 & u_2 & v_2 & u_3 & v_3 \end{array}\right)$$

$\left|e_1 \quad u_1 \quad v_1 e_2 \quad u_2 \quad v_2 e_3 \quad u_3 \quad v_3\right|=e_1\left|u_2 \quad v_2 u_3 \quad v_3\right|-e_2\left|u_1 \quad v_1 u_3 \quad v_3\right|$

$u \wedge v=-v \wedge u \quad(\lambda u+\mu w) \wedge v=\lambda(u \wedge v)+\mu(w \wedge v) ;(u \wedge v, w)=\mid u_1 \quad v_1$
$u \wedge v=0 \quad$ if and only if $u$ and $v$ are linearly dependent; (6.37)

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