# 微积分作业代写calclulus代考| Vector-valued functions

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## 微积分作业代写calclulus代考|The most general case f : Rn −→ Rm→ Rm

Although applications arise in more general cases of $\boldsymbol{f}: \mathbb{R}^{n} \longrightarrow \mathbb{R}^{m}$, we derive no benefit by specializing any further. We can instead reflect on the parallels

that may be drawn between “projections” of a more general scenario and the special cases we have already discussed.

A differentiable vector function $f: \mathbb{R}^{n} \longrightarrow \mathbb{R}^{m}$ with components $f_{i}, i=$ $1,2,3, \ldots, m$, which are real-valued functions of $\boldsymbol{x}$, is a vector-valued function
$$\boldsymbol{f}(\boldsymbol{x})=\left(f_{1}(\boldsymbol{x}), f_{2}(\boldsymbol{x}), f_{3}(\boldsymbol{x}), \ldots, f_{m}(\boldsymbol{x})\right)$$
When we say that $\boldsymbol{x} \in \mathbb{R}^{n}$ is in the domain of $\boldsymbol{f}$, where $R_{f} \subset \mathbb{R}^{m}$, we mean that $\boldsymbol{x}$ is in the domain of each component, the scalar functions $f_{i}, i=1,2, \ldots, m$. Then if we can assume that each of the $f_{i}$ is continuous and has continuous partial derivatives, we can compute the gradient for each component,
$$\nabla f_{i}(\boldsymbol{x})=\left(\frac{\partial f_{i}}{\partial x_{1}}, \ldots, \frac{\partial f_{i}}{\partial x_{n}}\right), i=1,2, \ldots, m .$$
We have met the gradient in Section 2.E, where it was used to determine the rate of change of a scalar function in a specified direction. So, for instance, for the case $n=2$ and $m=1$, we had $z=f(x, y)$ describing a surface in $3 \mathrm{D}$ space, and the rate of change of $z$ at a point $\left(x_{0}, y_{0}\right)$ in the direction of unit vector $\boldsymbol{u}=(u, v)$ was given by
$$D_{u} f\left(x_{0}, y_{0}\right)=\left.\nabla f\right|{0} \cdot \boldsymbol{u}=\left(\left.\frac{\partial f}{\partial x}\right|{0} \mathbf{i}+\left.\frac{\partial f}{\partial y}\right|{0} \mathbf{j}\right) \cdot(u \mathbf{i}+v \mathbf{j}),$$ or, in matrix notation, $$D{u} f\left(x_{0}, y_{0}\right)=\left[\left.\left.\frac{\partial f}{\partial x}\right|{0} \frac{\partial f}{\partial y}\right|{0}\right]\left(\begin{array}{l} u \ v \end{array}\right) .$$
With vector functions $\boldsymbol{f}(\boldsymbol{x})$ we have the potential to simultaneously find the rate of change of more that one scalar function using matrix multiplication. So for the case $n=2$ and $m=3$ we would have
$$\boldsymbol{f}\left(x_{0}, y_{0}\right)=\left[\begin{array}{ll} \left.\frac{\partial f_{1}}{\partial x}\right|{0} & \left.\frac{\partial f{1}}{\partial y}\right|{0} \ \left.\frac{\partial f{2}}{\partial x}\right|{0} & \left.\frac{\partial f{2}}{\partial y}\right|{0} \ \left.\frac{\partial f{3}}{\partial x}\right|{0} & \left.\frac{\partial f{3}}{\partial y}\right|_{0} \end{array}\right]\left(\begin{array}{c} u \ v \end{array}\right)$$
The $3 \times 2$ matrix on the right is an example of a Jacobian matrix. The Jacobian lies at the heart of every generalization of single-variable calculus to higher dimensions.

## 微积分作业代写calclulus代考|Remarks

• Before continuing, the reader might find it useful to revisit the discussions on coordinate systems and visualization of surfaces in Sections 1.D and 1.E.
• If we keep $u=u_{0}$ fixed we get $\boldsymbol{r}=\boldsymbol{r}\left(u_{0}, v\right)$, a vector function of one variable, $v$ (Figure 5.9). That is, restricting the variable $u$ results in a curve on $S$ called the constant $u$ curve. By the foregoing section this curve has a tangent vector given by
$$\boldsymbol{r}{v}^{\prime}\left(u{0}, v\right)=\frac{\partial \boldsymbol{r}}{\partial v}\left(u_{0}, v\right)=\left(\frac{\partial x}{\partial v}\left(u_{0}, v\right), \frac{\partial y}{\partial v}\left(u_{0}, v\right), \frac{\partial z}{\partial v}\left(u_{0}, v\right)\right)$$
• Similarly, if we keep $v=v_{0}$ fixed we get $\boldsymbol{r}=\boldsymbol{r}\left(u, v_{0}\right)$, a vector function of the single variable $u$. This too is a curve on $S$, called the constant $v$ curve. Analogously, this curve has a tangent vector given by
$$\boldsymbol{r}{u}^{\prime}\left(u, v{0}\right)=\frac{\partial \boldsymbol{r}}{\partial u}\left(u, v_{0}\right)=\left(\frac{\partial x}{\partial u}\left(u, v_{0}\right), \frac{\partial y}{\partial u}\left(u, v_{0}\right), \frac{\partial z}{\partial u}\left(u, v_{0}\right)\right)$$
• If $\boldsymbol{r}{u}^{\prime}\left(u{0}, v_{0}\right) \times \boldsymbol{r}{v}^{\prime}\left(u{0}, v_{0}\right) \neq 0$, which is the case for independent variables, then $\boldsymbol{r}{u}^{\prime} \times \boldsymbol{r}{v}^{\prime}$ is a vector normal to $S$ and normal to the tangent plane to $S$ (at the point $r\left(u_{0}, v_{0}\right)$ ) spanned by the vectors $r_{u}^{\prime}\left(u_{0}, v_{0}\right)$ and $\boldsymbol{r}{v}^{\prime}\left(u{0}, v_{0}\right)$.

## 微积分作业代写calclulus代考|The most general case f : Rn −→ Rm→ Rm

F(X)=(F1(X),F2(X),F3(X),…,F米(X))

∇F一世(X)=(∂F一世∂X1,…,∂F一世∂Xn),一世=1,2,…,米.

D你F(X0,和0)=∇F|0⋅你=(∂F∂X|0一世+∂F∂和|0j)⋅(你一世+vj),或者，在矩阵表示法中，D你F(X0,和0)=[∂F∂X|0∂F∂和|0](你 v).

F(X0,和0)=[∂F1∂X|0∂F1∂和|0 ∂F2∂X|0∂F2∂和|0 ∂F3∂X|0∂F3∂和|0](你 v)

## 微积分作业代写calclulus代考|Remarks

• 在继续之前，读者可能会发现重温第 1.D 节和第 1.E 节中关于坐标系和表面可视化的讨论很有用。
• 如果我们保持你=你0固定我们得到r=r(你0,v)，一个变量的向量函数，v（图 5.9）。也就是限制变量你结果是一条曲线小号称为常数你曲线。通过前面的部分，这条曲线的切向量由下式给出
rv′(你0,v)=∂r∂v(你0,v)=(∂X∂v(你0,v),∂和∂v(你0,v),∂和∂v(你0,v))
• 同样，如果我们保持v=v0固定我们得到r=r(你,v0), 单变量的向量函数你. 这也是一条曲线小号，称为常数v曲线。类似地，这条曲线的切向量由下式给出
r你′(你,v0)=∂r∂你(你,v0)=(∂X∂你(你,v0),∂和∂你(你,v0),∂和∂你(你,v0))
• 如果r你′(你0,v0)×rv′(你0,v0)≠0，对于自变量就是这种情况，那么r你′×rv′是一个垂直于小号并垂直于切平面小号（此时r(你0,v0)) 由向量跨越r你′(你0,v0)和rv′(你0,v0).