벡터와 공간 전체 문제

The distance from $P$ to the point $(0,0,1)$ is $\sqrt{x^2+y^2+(z-1)^2}$. The distance from $P$ to the specified plane is $|z+1|$. Hence the equation of the surface is

$$x^2+y^2+(z-1)^2=(z+1)^2\text{ or } x^2+y^2=4z$$

All points on this surface have $z\ge 0$. The set of points on the surface that have any fixed value, $z_0\ge 0$, of $z$ consists of a circle that is centred on the $z$–axis, is parallel to the $xy$-plane and has radius $2\sqrt{z_0}$. The surface consists of a stack of these circles, starting with a point at the origin and with radius increasing vertically. The surface is a paraboloid and is sketched below.

(a) For each fixed $c>0$, the level curve $x^2+2y^2=c$ is the ellipse centred on the origin with $x$ semi axis $\sqrt{c}$ and $y$ semi axis $\sqrt{c/2}$. If $c=0$, the level curve $x^2+2y^2=c=0$ is the single point $(0,0)$.

(b) For each fixed $c\ne 0$, the level curve $xy=c$ is a hyperbola centred on the origin with asymptotes the $x$- and $y$-axes. If $c>0$, any $x$ and $y$ obeying $xy=c>0$ are of the same sign. So the hyperbola is contained in the first and third quadrants. If $c<0$, any $x$ and $y$ obeying $xy=c>0$ are of opposite sign. So the hyperbola is contained in the second and fourth quadrants. If $c=0$, the level curve $xy=c=0$ is the single point $(0,0)$.

(c) For each fixed $c\ne 0$, the level curve $xe^{-y}=c$ is the logarithmic curve $y=-\ln\frac{c}{x}$. Note that, for $c>0$, the curve

• is restricted to $x>0$, so that $\frac{c}{x}>0$ and $\ln \frac{c}{x}$ is defined, and that
• as $x\rightarrow 0^+$, $y$ goes to $-\infty$, while
• as $x\rightarrow +\infty$, $y$ goes to $+\infty$, and
• the curve crosses the $x$-axis (i.e. has $y=0$) when $x=c$.

and for $c<0$, the curve

• is restricted to $x<0$, so that $\frac{c}{x}>0$ and $\ln \frac{c}{x}$ is defined, and that
• as $x\rightarrow 0^-$, $y$ goes to $-\infty$, while
• as $x\rightarrow -\infty$, $y$ goes to $+\infty$, and
• the curve crosses the $x$-axis (i.e. has $y=0$) when $x=c$.

If $c=0$, the level curve $xe^{-y}=c=0$ is the $y$-axis, $x=0$.

If $C=0$, the level curve $f=C=0$ is just the line $y=0$. If $C\ne 0$ (of either sign), we may rewrite the equation, $f(x,y)=\frac{2y}{x^2+y^2}=C$, of the level curve $f=C$ as

$$x^2-\frac{2}{C}y+y^2=0 \iff x^2+\left(y-\frac{1}{C}\right)^2 =\frac{1}{C^2}$$

which is the equation of the circle of radius $\frac{1}{|C|}$ centred on $\left(0\,,\,\frac{1}{C}\right)$.

Remark. To be picky, the function $f(x,y)=\frac{2y}{x^2+y^2}$ is not defined at $(x,y)=(0,0)$. The question should have either specified that the domain of $f$ excludes $(0,0)$ or have specified a value for $f(0,0)$. In fact, it is impossible to assign a value to $f(0,0)$ in such a way that $f(x,y)$ is continuous at $(0,0)$, because $\lim_{x\rightarrow 0}f(x,0)=0$ while $\lim_{y\rightarrow 0}f(0,|y|)=\infty$. So it makes more sense to have the domain of $f$ being $\mathbb{R}^2$ with the point $(0,0)$ removed. That's why there is a little hole at the origin in the above sketch.

(a) We can rewrite the equation as

$$x^2 + y^2 = (z-1)^2 - 1$$

The right hand side is negative for $|z-1|<1$, i.e. for $0<z<2$. So no point on the surface has $0<z<2$. For any fixed $z$, outside that range, the curve $x^2 + y^2 = (z-1)^2 - 1$ is the circle of radius $\sqrt{(z-1)^2 - 1}$ centred on the $z$–axis. That radius is $0$ when $z=0,2$ and increases as $z$ moves away from $z=0,2$. For very large $|z|$, the radius increases roughly linearly with $|z|$. Here is a sketch of some level curves.

(b) The surface consists of two stacks of circles. One stack starts with radius $0$ at $z=2$. The radius increases as $z$ increases. The other stack starts with radius $0$ at $z=0$. The radius increases as $z$ decreases. This surface is a hyperboloid of two sheets. Here are two sketchs. The sketch on the left is of the part of the surface in the first octant. The sketch on the right of the full surface.

For each fixed $z$, $4x^2+y^2=1+z^2$ is an ellipse. So the surface consists of a stack of ellipses one on top of the other. The semi axes are $\frac{1}{2}\sqrt{1+z^2}$ and $\sqrt{1+z^2}$. These are smallest when $z=0$ (i.e. for the ellipse in the $xy$-plane) and increase as $|z|$ increases. The intersection of the surface with the $xz$-plane (i.e. with the plane $y=0$) is the hyperbola $4x^2-z^2=1$ and the intersection with the $yz$-pane (i.e. with the plane $x=0$) is the hyperbola $y^2-z^2=1$. Here are two sketches of the surface. The sketch on the left only shows the part of the surface in the first octant (with axes).

(a) The graph is $z=\sin x$ with $(x,y)$ running over $0\le x\le 2\pi$, $0\le y\le 1$. For each fixed $y_0$ between $0$ and $1$, the intersection of this graph with the vertical plane $y=y_0$ is the same sin graph $z=\sin x$ with $x$ running from $0$ to $2\pi$. So the whole graph is just a bunch of 2-d sin graphs stacked side-by-side. This gives the graph on the left below.

(b) The graph is $z=\sqrt{x^2+y^2}$. For each fixed $z_0\ge 0$, the intersection of this graph with the horizontal plane $z=z_0$ is the circle $\sqrt{x^2+y^2}=z_0$. This circle is centred on the $z$-axis and has radius $z_0$. So the graph is the upper half of a cone. It is the sketch on the right above.

(c) The graph is $z=|x|+|y|$. For each fixed $z_0\ge 0$, the intersection of this graph with the horizontal plane $z=z_0$ is the square $|x|+|y|=z_0$. The side of the square with $x,y\ge 0$ is the straight line $x+y=z_0$. The side of the square with $x\ge 0$ and $y\le 0$ is the straight line $x-y=z_0$ and so on. The four corners of the square are $(\pm z_0,0,z_0)$ and $(0, \pm z_0,z_0)$. So the graph is a stack of squares. It is an upside down four-sided pyramid. The part of the pyramid in the first octant (that is, $x,y,z\ge 0$) is the sketch below.

(a) For each fixed $z_0$, the $z=z_0$ cross-section (parallel to the $xy$-plane) of this surface is an ellipse centered on the origin with one semiaxis of length 2 along the $x$-axis and one semiaxis of length 4 along the $y$-axis. So this is an elliptic cylinder parallel to the $z$-axis. Here is a sketch of the part of the surface above the $xy$–plane.

(b) This is a plane through $(4,0,0)$, $(0,4,0)$ and $(0,0,2)$. Here is a sketch of the part of the plane in the first octant.

(c) For each fixed $x_0$, the $x=x_0$ cross-section parallel to the $yz$-plane is an ellipse with semiaxes $3\sqrt{1+\frac{x_0^2}{16}}$ parallel to the $y$-axis and $2\sqrt{1+\frac{x_0^2}{16}}$ parallel to the $z$-axis. As you move out along the $x$-axis, away from $x=0$, the ellipses grow at a rate proportional to $\sqrt{1+\frac{x^2}{16}}$, which for large $x$ is approximately $\frac{|x|}{4}$. This is called a hyperboloid of one sheet. Its

(d) For each fixed $y_0$, the $y=x_0$ cross-section (parallel to the $xz$-plane) is a circle of radius $|y|$ centred on the $y$-axis. When $y_0=0$ the radius is $0$. As you move further from the $xz$-plane, in either direction, i.e. as $|y_0|$ increases, the radius grows linearly. The full surface consists of a bunch of these circles stacked sideways. This is a circular cone centred on the $y$-axis.

(e) This is an ellipsoid centered on the origin with semiaxes $3$, $\sqrt{12}=2\sqrt{3}$ and $3$ along the $x$, $y$ and $z$-axes, respectively.

(f) Completing three squares, we have that $x^2+y^2+z^2+4x-by+9z-b=0$ if and only if $(x+2)^2+\big(y-\frac{b}{2}\big)^2+\big(z+\frac{9}{2}\big)^2 =b+4+\frac{b^2}{4}+\frac{81}{4}$. This is a sphere of radius $r_b=\frac{1}{2}\sqrt{b^2+4b+97}$ centered on $\frac{1}{2}(-4,b,-9)$.

(g) There are no points on the surface with $x<0$. For each fixed $x_0>0$ the cross-section $x=x_0$ parallel to the $yz$-plane is an ellipse centred on the $x$–axis with semiaxes $\sqrt{x_0}$ in the $y$-axis direction and $\frac{3}{2}\sqrt{x_0}$ in the $z$–axis direction. As you increase $x_0$, i.e. move out along the $x$-axis, the ellipses grow at a rate proportional to $\sqrt{x_0}$. This is an elliptic paraboloid with axis the $x$-axis.

(h) This is called a parabolic cylinder. For any fixed $y_0$, the $y=y_0$ cross-section (parallel to the $xz$-plane) is the upward opening parabola $z=x^2$ which has vertex on the $y$-axis.