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Applet loadingA graph of the elliptic paraboloid $f(x,y)=-x^2-2y^2$ over the domain $-2 \le x \le 2$ and $-2 \le y \le 2$.

The applet was made using three.js and requires Javascript as well as a browser that supports WebGL.

The animations below show the normal vector fields in black as the point out of the surface. The unit vectors with respect to spherical coordinates generate a rather neat vector field which I'd rather not plot by hand. The function f takes two inputs, $x$ and $y$, and returns a single number, which we call $z$. If you choose a point $(x,y)$ in the $xy$-plane, then $z=f(x,y)$ represents the height of the graph at that point.

If you fix either $x$ or $y$ and increase the other variable, the function will increase with a slope of one. The slopes of the plane are hard to visualize as the $z$-axis is scaled differently than the $x$- and $y$-axes.

The Java applet did not load, and the above is only a static image representing one view of the applet. This page is derived from parts of the Mathematica notebook Graphing Surfaces, written by Jonathan Rogness for the University of Minnesota course Math 2374.

Graph of the elliptic paraboloid $z=x^2+y^2$ plotted on the circular domain $x^2+y^2 \le 2$.

Image: Example for area of region by double integralExample region for calculating the area of a region using a double integral.

Image: Change order of integration example region with exponential, x firstExample region illustrating process of changing the order of integration in double integrals, shown with x as the inner integral. Image: Change order of integration example region with exponential, y firstExample region illustrating process of changing the order of integration in double integrals, shown with y as the inner integral.

Image: Change order of integration example region with sinusoid, x firstExample region illustrating process of changing the order of integration in double integrals, shown with x as the inner integral.

Image: Change order of integration example region with sinusoid, y firstExample sinusoidal region illustrating process of changing the order of integration in double integrals, shown with y as the inner integral.

Image: Change order of integration example triangular regionExample region illustrating process of changing the order of integration in double integrals. Image: Example region for changing variables in a double integralExample region for a general change of variables in a double integral.

Image: Example region for changing variables in a double integralExample region for evaluating a double integral by changing variables to polar coordinates. Image: Double integral over triangular region, integrating x firstA triangular region described in a way to facilitate integrating with respect to x first. Image: Double integral over triangular region, integrating y firstA triangular region described in a way to facilitate integrating with respect to y first.

Image: Double integral region where best to integrate x firstA region where it's easier to integrate with respect to x first.

Image: Double integral region where best to integrate x first with boundaries labeledA region with boundaries labeled, demonstrating it's easier to integrate with respect to x first.

Image: Double integral region where best to integrate x first, not yA region where it's easy to integrate with respect to x first but hard to integrate with respect to y first. Image: Double integral as volume under a surfaceThe double integral can be interpreted as the volume under a surface. Image: Double integral as volume under a surface, with box illustrating Riemann sumThe double integral can be interpreted as based on a Riemann sum estimating the volume under a surface. Image: Elliptic paraboloid level curvesThe level curves of the elliptic paraboloid are concentric ellipses.

Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License.

The idea of feeding an interesting planar curve into a graph or parametrization in three space allows us to create many interesting examples. Notice that A=0 gives the coordinate planes whereas the graphs for A0 are found in differing octants. The graph of the function $g(x,y)=1$ is the horizontal plane of height 1.More information about applet. The graph of the function $g(x,y)=x+y$ is shown with all axis at the same scale to better reveal the steepness of the angled plane.More information about applet.

The graph of the function $f(x,y)=x^2+y^2$ is called an elliptic paraboloid.More information about applet.

The colored line indicates the limit of g(t) formed by composing f with a linear path through (0,0).

The colored line indicates the limit of g(t) as t --> 0 where g is formed by composing f with a linear path through (0,0). I have no name for this, in fact, generally it might be interesting to investigate what graphs of the form F(x,y,z)=k for F a trilinear form. You can see that the values for differing linear paths differs hence the limit at the origin for f(x,y) does not exist. You can see that the values for differing linear paths all agree since they cut through the origin.

The applet was made using three.js and requires Javascript as well as a browser that supports WebGL.

The animations below show the normal vector fields in black as the point out of the surface. The unit vectors with respect to spherical coordinates generate a rather neat vector field which I'd rather not plot by hand. The function f takes two inputs, $x$ and $y$, and returns a single number, which we call $z$. If you choose a point $(x,y)$ in the $xy$-plane, then $z=f(x,y)$ represents the height of the graph at that point.

If you fix either $x$ or $y$ and increase the other variable, the function will increase with a slope of one. The slopes of the plane are hard to visualize as the $z$-axis is scaled differently than the $x$- and $y$-axes.

The Java applet did not load, and the above is only a static image representing one view of the applet. This page is derived from parts of the Mathematica notebook Graphing Surfaces, written by Jonathan Rogness for the University of Minnesota course Math 2374.

Graph of the elliptic paraboloid $z=x^2+y^2$ plotted on the circular domain $x^2+y^2 \le 2$.

Image: Example for area of region by double integralExample region for calculating the area of a region using a double integral.

Image: Change order of integration example region with exponential, x firstExample region illustrating process of changing the order of integration in double integrals, shown with x as the inner integral. Image: Change order of integration example region with exponential, y firstExample region illustrating process of changing the order of integration in double integrals, shown with y as the inner integral.

Image: Change order of integration example region with sinusoid, x firstExample region illustrating process of changing the order of integration in double integrals, shown with x as the inner integral.

Image: Change order of integration example region with sinusoid, y firstExample sinusoidal region illustrating process of changing the order of integration in double integrals, shown with y as the inner integral.

Image: Change order of integration example triangular regionExample region illustrating process of changing the order of integration in double integrals. Image: Example region for changing variables in a double integralExample region for a general change of variables in a double integral.

Image: Example region for changing variables in a double integralExample region for evaluating a double integral by changing variables to polar coordinates. Image: Double integral over triangular region, integrating x firstA triangular region described in a way to facilitate integrating with respect to x first. Image: Double integral over triangular region, integrating y firstA triangular region described in a way to facilitate integrating with respect to y first.

Image: Double integral region where best to integrate x firstA region where it's easier to integrate with respect to x first.

Image: Double integral region where best to integrate x first with boundaries labeledA region with boundaries labeled, demonstrating it's easier to integrate with respect to x first.

Image: Double integral region where best to integrate x first, not yA region where it's easy to integrate with respect to x first but hard to integrate with respect to y first. Image: Double integral as volume under a surfaceThe double integral can be interpreted as the volume under a surface. Image: Double integral as volume under a surface, with box illustrating Riemann sumThe double integral can be interpreted as based on a Riemann sum estimating the volume under a surface. Image: Elliptic paraboloid level curvesThe level curves of the elliptic paraboloid are concentric ellipses.

Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License.

The idea of feeding an interesting planar curve into a graph or parametrization in three space allows us to create many interesting examples. Notice that A=0 gives the coordinate planes whereas the graphs for A0 are found in differing octants. The graph of the function $g(x,y)=1$ is the horizontal plane of height 1.More information about applet. The graph of the function $g(x,y)=x+y$ is shown with all axis at the same scale to better reveal the steepness of the angled plane.More information about applet.

The graph of the function $f(x,y)=x^2+y^2$ is called an elliptic paraboloid.More information about applet.

The colored line indicates the limit of g(t) formed by composing f with a linear path through (0,0).

The colored line indicates the limit of g(t) as t --> 0 where g is formed by composing f with a linear path through (0,0). I have no name for this, in fact, generally it might be interesting to investigate what graphs of the form F(x,y,z)=k for F a trilinear form. You can see that the values for differing linear paths differs hence the limit at the origin for f(x,y) does not exist. You can see that the values for differing linear paths all agree since they cut through the origin.

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