这是一份leeds利兹大学MATH5113M01作业代写的成功案例

• Let $Y \rightarrow X$ be a vector bundle with a typical fibre $V$. By $Y^{} \rightarrow X$ is denoted the dual vector bundle with the typical fibre $V^{}$ dual of $V$. The interior product of $Y$ and $Y^{}$ is defined as a fibred morphism $$ J: Y \otimes Y^{} \underset{X}{\longrightarrow} X \times \mathbb{R} .
  $$
• Let $Y \rightarrow X$ and $Y^{\prime} \rightarrow X$ be vector bundles with typical fibres $V$ and $V^{\prime}$, respectively. Their Whitney sum $Y \underset{X}{\oplus} Y^{\prime}$ is a vector bundle over $X$ with the typical fibre $V \oplus V^{\prime}$.
• Let $Y \rightarrow X$ and $Y^{\prime} \rightarrow X$ be vector bundles with typical fibres $V$ and $V^{\prime}$, respectively. Their tensor product $Y \otimes Y^{\prime}$ is a vector bundle over $X$ with the typical fibre $V \otimes V^{\prime}$. Similarly, the exterior product of vector bundles $Y \underset{X}{\wedge} Y^{\prime}$ is defined. The exterior product
  is called the exterior bundle

MATH5113M01 COURSE NOTES ：

Vector fields on a manifold $Z$ are global sections of the tangent bundle $T Z \rightarrow Z$.

The set $\mathcal{T}(Z)$ of vector fields on $Z$ is both a $C^{\infty}(Z)$-module and a real Lie algebra with respect to the Lie bracket
$$
\begin{aligned}
&u=u^{\lambda} \partial_{\lambda}, \quad v=v^{\lambda} \partial_{\lambda} \
&{[v, u]=\left(v^{\lambda} \partial_{\lambda} u^{\mu}-u^{\lambda} \partial_{\lambda} v^{\mu}\right) \partial_{\mu^{-}}}
\end{aligned}
$$
Given a vector field $u$ on $X$, a curve
$$
c: \mathbb{R} D(,) \rightarrow Z
$$