高数期末

曲率（参数 $t$ 定义的三维曲线）公式 $$ \kappa(t) = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3} $$

其在二维的特例 $r(t) = (r \cos t, r \sin t, 0)$，代入即可得 $\frac{1}{r}$。

这个 test 可能要记一下。

Theorem: Second Partials Test

Suppose that $f(x,y)$ has continuous second partial derivatives in a neighborhood of $(x_0,y_0)$ and that $\nabla f(x_0,y_0)=0$. Let

$$ D=D(x_0,y_0)=f_{xx}(x_0,y_0)f_{yy}(x_0,y_0)-f_{xy}^2(x_0,y_0) $$ Then:

If $D>0$ and $f_{xx}(x_0,y_0)<0$, then $f(x_0,y_0)$ is a local maximum value.
If $D>0$ and $f_{xx}(x_0,y_0)>0$, then $f(x_0,y_0)$ is a local minimum value.
If $D<0$, then $f(x_0,y_0)$ is not an extreme value (i.e., $(x_0,y_0)$ is a saddle point).
If $D=0$, then the test is inconclusive.

往届考题选做¶

Problem 22 (4 points)¶

Let $f(x,y)$ be continuous on its domain, and
$$ f(x,y) = e^{x^2+y^2} + xy \iint_D xy \, f(x,y) dA $$
where $D = {(x, y) : 0 \leq x \leq 1,: 0 \leq y \leq 1}$. Find $f(x,y)$.

和高数1一个套路，后面那串积分是个定值，设其为 $C$ 得到 $f(x,y)$，然后代回这个积分，从而得到关于 $C$ 的方程，解出即可。

Problem 21 (11 points)¶

Assume that $\mathbf{F}(x, y) = (2xy + y)\mathbf{i} + (x^2 + x + 1)\mathbf{j}$.

(1) (3 points) Show that $\mathbf{F}(x,y)$ is conservative.

conservative 大概是说其在任意闭曲线上的积分为 $0$。由 Green 公式，即 $$ \nabla \times F \equiv 0 $$ 简单计算即可

(2) (4 points) Find $f$ such that $\mathbf{F} = \nabla f$.

先从 $f_x$ 反推出待定一个 $C(y)$ 的 $f$，然后再考虑 $f_y$ 求出 $C(y)$。

(3) (4 points) Calculate $\int_C (2xy + y) dx + (x^2 + x + 1) dy$, where $C$ is any path from (0,0) to (2,1).

其等于二问中 $f(2,1) - f(0,0)$。

Problem 20¶

Evaluate $\iint_G xyz ds$, where $G$ is the part of the plane $z = 2x + 3y$ above the triangular region with vertices (0,0,0), (1,0,0) and (1,1,0).

把 $ds$ 投影到 xy 平面内 $$ \sqrt{z_x^2+z_y^2+1}dxdy $$ 即可。

Problem 19¶

Calculate $\oint_C 2y dx - 2x dy$, where $C$ is the boundary of the triangle with vertices (0,0), (2,1) and (0,1).

套 Green 公式即转化为求三角形面积乘一个常数。