[b][size=150]Some simple surfaces in 3D space[/size][/b][br][br]Besides planes, there are some simple surfaces in [math]\mathbb{R}^3[/math] that can be described by simple equations. [br][br][u]Cylinders[/u][br][br]Suppose [math]C[/math] is a closed curve on a plane. A [b]cylinder[/b] can be generated by projecting [math]C[/math] along the axis perpendicular to the plane. For example, if the closed curve [math]C[/math] is a unit circle on the xy-plane. Then the equation [br][br][math]x^2+y^2=1[/math][br][br]describe the cylinder of [math]C[/math] (a circular cylinder) parallel to z-axis because for any cross-section at [math]z[/math], [math](x,y,z)[/math] is on the cylinder as long as [math]x^2+y^2=1[/math] i.e. [math]x[/math] and [math]y[/math] are independent of [math]z[/math]. Hence [math]z[/math] does not appear in the equation.[br][br]In the applet below, draw the cylinder of the circle on the yz-plane centered at [math](0,0,0)[/math] with radius [math]2[/math] parallel to x-axis.[br][br][u]Spheres[/u][br][br]The equation for the sphere centered at [math](x_0,y_0,z_0)[/math] with radius [math]r>0[/math] is clearly as follows:[br][br][math](x-x_0)^2+(y-y_0)^2+(z-z_0)^2=r^2[/math][br][br]The above equation describes the fact that the distance between any point [math](x,y,z)[/math] and [math](x_0,y_0,z_0)[/math] equals [math]r[/math].[br][br]If we compress/stretch a sphere along the three axes by some factors, we will get a more general kind of surfaces called [b]ellipsoid[/b]. For example, you can draw the ellipsoid [math]x^2+\frac{y^2}{4}+\frac{z^2}9=1[/math] in the applet below.[br][br][u]Cones[/u][br][br]Consider the right circular cone with apex at the origin [math](0,0,0)[/math] such that it is open upwards and its semi-vertical angle is [math]\frac{\pi}4[/math]. The equation of the cone is as follows:[br][br][math]z=\sqrt{x^2+y^2}[/math][br][br]If we want to include the cone below the xy-plane, we can use the following equation:[br][br][math]z^2=x^2+y^2[/math][br][br]More generally, if the semi-vertical angle is [math]\theta[/math], the equation of the cone becomes[br][br][math]cz=\sqrt{x^2+y^2}[/math][br][br]where [math]c=\tan \theta[/math].[br][br]You can draw a cone in the applet below.[br]

[b][size=150]Multivariable functions[/size][/b][br][br]In this chapter, we mainly study [b]multivariable functions[/b] i.e. real-valued functions of more than one variable. Suppose [math]f[/math] is a real-valued function of two variables. We can express this function as follows:[br][br][math]z = f(x,y)[/math][br][br]where [math](x, y)[/math] is any point in the [b]domain[/b] [math]D[/math] of [math]f[/math], a subset of [math]\mathbb{R}^2[/math] containing all the possible inputs of the function. Then a unique real number [math]z[/math] is assigned by the function for each [math](x,y)[/math]. The [b]range[/b] of [math]f[/math] is the set of all possible outputs of the function i.e. the set of all possible values of [math]z[/math].[br][br][u]Example[/u]: Suppose [math]z=f(x,y)=\sqrt{4-x^2-y^2}[/math]. It domain is [math]\left\{(x,y) \ | \ x^2+y^2\leq 4\right\}[/math] and its range is [math]\left\{z \ | \ 0\leq z \leq 2\right\}[/math].[br][br]Similarly, we can also consider a function of three variables [math]w=f(x,y,z)[/math], or in general, a function of [math]n[/math] variables [math]y=f(x_1,x_2,\ldots, x_n)[/math]. [br][br]As we know, we can visualize a function of one variable [math]y=f(x)[/math] as a graph in [math]\mathbb{R}^2[/math] i.e. it is a set of all points of the form [math](x,f(x))[/math]. Similarly, we can visualize a function of two variable [math]z=f(x,y)[/math] as a graph in [math]\mathbb{R}^3[/math] i.e. it is a set of all points of the form [math](x,y,f(x,y))[/math].[br][br]In the applet below, you can draw the graph of [math]z=f(x,y)[/math] and then find the point [math]P[/math] on the graph corresponding to the point [math]A=(x,y)[/math].[br][br]You can try the following functions:[br][br][list][*][math]f(x,y)=1-x-\frac 12 y [/math][/*][br][*][math]h(x,y)=x^2+y^2 [/math][/*][br][*][math]g(x,y)=\sqrt{1+x^2+y^2} [/math][/*][br][/list]

[b][size=150]Level curves[/size][/b][br][br]Consider the graph of [math]z=f(x,y)[/math]. The [b]contour curve[/b] is the intersection of the graph and the horizontal plane [math]z=k[/math] for some real number [math]k[/math]. The [b]level curve[/b] is the projection of the contour curve onto xy-plane. It can be represented by the equation [math]f(x,y)=k[/math]. [br][br]In the applet below, the level curve [math]f(x,y)=k[/math] is shown in the left panel and the corresponding contour curve is shown in the right panel. You can view the level curves for the following functions:[br][list][*] [math]f(x,y)=\sqrt{x^2+y^2-1} [/math][/*][br][*] [math]g(x,y)=\frac yx [/math][/*][/list][br][br]Moreover, you can use the applet to draw multiple level curves (e.g. [math]f(x,y)=k[/math] for [math]k=0,1,2,3,4,5[/math] ) to form a [b]contour map[/b] of the graph using the following method:[br][br]First, enter the command "k=slider(0,5,1)". Then right-click on the level curve and check the box "Show trace". Drag the slider to generate the level curves for [math]k=0,1,2,3,4,5[/math].[br][br]

[b][size=150]Level surfaces[/size][/b][br][br]For functions of three variables [math]w=f(x,y,z)[/math], it is not easy to visualize its graph since it is a three-dimensional object in four-dimensional space. However, we can visualize the intersection between the graph and [math]w=w_0[/math] as it is a two-dimensional object i.e. a surface. It is called the [b]level surface[/b] of [math]f[/math]. Its equation is [math]f(x,y,z)=w_0[/math].[br][br][u]Example[/u]: Let [math]w=f(x,y,z)=x^2+y^2+z^2[/math]. For [math]w=w_0>0[/math], its level surface has the equation [math]x^2+y^2+z^2=w_0[/math] i.e. a sphere centered at the origin with radius [math]\sqrt{w_0}[/math].[br][br]In the applet below, you can visualize the level surface [math]f(x,y,z)=k[/math] for various functions.[br][br](Note: GeoGebra can only handle the plotting of level surface for quadratic polynomials in three variables.)[br]