Applied mathematics is the process of applying mathematics to real world problems, and also learning and developing mathematical tools to solve these problems. Many of the advances in mathematics, such as in calculus, differential equations, vector calculus, Fourier analysis (which you will cover in the Second Year), computation and geometry have come directly from studying such applications.

这是一份Bath巴斯大学学MA10236作业代写的成功案

Let $\Sigma$ be a surface in $\mathbf{R}^{3}$ and let $\mathbf{f}(x, y, z)=f_{1}(x, y, z) \mathbf{i}+f_{2}(x, y, z) \mathbf{j}+$ $f_{s}(x, y, z) \mathbf{k}$ be a vector field defined on some subset of $\mathbf{R}^{3}$ that contains $\Sigma$. The surface integral of $f$ over $\Sigma$ is
$$
\iint_{\Sigma} \mathbf{f} \cdot d \sigma=\iint_{\Sigma} \mathbf{f} \cdot \mathbf{n} d \sigma \text {, }
$$
where, at any point on $\Sigma, \mathbf{n}$ is the outward unit normal vector to $\Sigma$.

MA10236  COURSE NOTES ：

(Divergence Theorem) Let $\Sigma$ be a closed surface in $\mathbf{R}^{3}$ which bounds a solid $S$, and let $\mathbf{f}(x, y, z)=f_{1}(x, y, z) \mathbf{i}+f_{2}(x, y, z) \mathbf{j}+f_{3}(x, y, z) \mathbf{k}$ be a vector field defined on some subset of $\mathbb{R}^{3}$ that contains $\Sigma$. Then
$$
\iint_{\Sigma} \mathbf{f} \cdot d \sigma=\iiint_{S} \operatorname{div} \mathbf{f} d V
$$
where
$$
\operatorname{div} \mathbf{f}=\frac{\partial f_{1}}{\partial x}+\frac{\partial f_{2}}{\partial y}+\frac{\partial f_{3}}{\partial z}
$$
is called the divergence of $f$.