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# 数学代写|PDE代写Partial Differential Equations代考|MAP4341/5345 Line Integral

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## 数学代写|PDE代写Partial Differential Equations代考|Line Integral

To start with, suppose a force field $f=f(\mathbf{r})$ acts along a smooth curve $C:=C_\gamma \subset$ $\mathbb{R}^n$, say parametrised by a smooth function $\gamma:[a, b] \rightarrow \mathbb{R}^n$, where the position vector $\mathbf{r}=\mathbf{r}(t)$ of a moving point along $C$ is given by
$$\mathbf{r}(t):=\left(\gamma_1(t), \ldots, \gamma_n(t)\right), \quad t \in I=[a, b],$$
such that $\left|\mathbf{r}^{\prime}(t)\right|>0$ for all $t$. The work done by the force $f$ in displacing a particle through a line element $d \ell$ along the curve $C$ is given by $f \cdot d \ell$. So, the total work done $W$ by the force $f$ in moving along the curve $C$ is given by the line integral
$$W:=\int_C f \cdot d \ell=\int_a^b\left(f \cdot \mathbf{r}^{\prime}\right) d t$$
The expression $\boldsymbol{f} \cdot d \ell$ is also known as a 1 -form field. Clearly, the value of the line integral depends on the path taken to move between points (Theorem 2.23), and also on the preferred orientation of the underlying parametrised curve. A similar remark holds for the other two types of line integrals as given below.
$$\int_C \varphi d \ell \text { and } \int_C F \times d \ell$$
The orientation ${ }^3$ of a parametrised curve also plays a significant role while finding a potential function of a vector field.

## 数学代写|PDE代写Partial Differential Equations代考|Flux and Divergence Theorem

As it has been for the line integral of a vector field along a smooth curve, the surface integral (or the volume integral) of a vector field over a regular surface $S$ also depends on the orientation of the surface. To introduce the concept, let $\mathbf{r}=\mathbf{r}(u, v): \Omega \rightarrow$ $\mathbb{R}^3$ be a parametrisation of $S$. In general, we study the geometry of $S$ at a point $a=\left(u_0, v_0\right) \in \Omega$ by using the two (orthogonal) curves given by
$$\mathbf{r}_1(u)=\mathbf{r}\left(u, v_0\right) \quad \text { and } \quad \mathbf{r}_2(v)=\mathbf{r}\left(u_0, v\right),$$
respectively, called the $u$-curve and $v$-curve. Notice that the derivatives $\mathbf{r}_u=\mathbf{r}^{\prime}(u)$ and $\mathbf{r}_v=\mathbf{r}^{\prime}(v)$ are, respectively, the tangent vectors to the two curves $\mathbf{r}_1$ and $\mathbf{r}_2$ on S. Also, by the vector identity
$$\left|\mathbf{r}_u \times \mathbf{r}_v\right|^2=\left(\mathbf{r}_u \times \mathbf{r}_v\right) \cdot\left(\mathbf{r}_u \times \mathbf{r}_v\right)=\operatorname{det}\left(\begin{array}{l} \mathbf{r}_u \cdot \mathbf{r}_u \mathbf{r}_u \cdot \mathbf{r}_v \ \mathbf{r}_v \cdot \mathbf{r}_u \mathbf{r}_v \cdot \mathbf{r}_v \end{array}\right)$$
it follows that the regularity condition as given in Definition $2.27$ is equivalent to the condition that the vectors $\mathbf{r}_u, \mathbf{r}_v$ are linearly independent. Therefore, there is a unique (shifted) tangent plane $\Pi(a)$ at $\mathbf{r}(\boldsymbol{a}) \in S$ spanned by the tangent vectors $\mathbf{r}_u$ and $\mathbf{r}_v$. In fact, the two vectors form a natural basis for the tangent plane $\Pi(\boldsymbol{x})$ in the sense we explain shortly. In particular, if $\varphi \in C^1(\Omega)$, then for the regular surface

$$\Gamma_{\varphi}(x, y, z): \quad \varphi(x, y)-z=0$$
we have $\mathbf{r}x=\left(1,0, \varphi_x\right)$ and $\mathbf{r}_y=\left(0,1, \varphi_y\right)$, which implies that $$\mathbf{r}_x \times \mathbf{r}_y=\left(-\varphi_x,-\varphi_y, 1\right),$$ and the equation of the tangent plane at a point $\boldsymbol{a}=\left(x_0, y_0, z_0\right)$ is given by $\varphi_x\left(x-x_0\right)+\varphi_y\left(y-y_0\right)-\left(z-z_0\right)=0, \quad$ where $z_0=\varphi\left(x_0, y_0\right) .$ Therefore, for any $x \in \Gamma{\varphi}$, we have
$$\mathbf{n}(\boldsymbol{x})=\pm \frac{\mathbf{r}_u \times \mathbf{r}_v}{\left|\mathbf{r}_u \times \mathbf{r}_v\right|}=\frac{\left(-\varphi_x,-\varphi_y, 1\right)}{\sqrt{\varphi_x^2+\varphi_y^2+1}}$$

## 数学代写|PDE代写Partial Differential Equations代考|Line Integral

$$\mathbf{r}(t):=\left(\gamma_1(t), \ldots, \gamma_n(t)\right), \quad t \in I=[a, b],$$

$$W:=\int_C f \cdot d \ell=\int_a^b\left(f \cdot \mathbf{r}^{\prime}\right) d t$$

$$\int_C \varphi d \ell \text { and } \int_C F \times d \ell$$

## 数学代写|PDE代写Partial Differential Equations代考|Flux and Divergence Theorem

$$\mathbf{r}_x \times \mathbf{r}_y=\left(-\varphi_x,-\varphi_y, 1\right),$$

$$\mathbf{n}(\boldsymbol{x})=\pm \frac{\mathbf{r}_u \times \mathbf{r}_v}{\left|\mathbf{r}_u \times \mathbf{r}_v\right|}=\frac{\left(-\varphi_x,-\varphi_y, 1\right)}{\sqrt{\varphi_x^2+\varphi_y^2+1}}$$

## MATLAB代写

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