Section II.1: Finding Limits of Integrals in Two Dimensions: A Case Study
Second Section Materials
Applet on multiple integration
in two dimensions with Cartesian coordinates This applet lets you specify a function to be integrated, and up to three boundary functions in two variables. It the chops the region in the graph into little square, The little square that are entirely in the region are graphed in dark blue, and those that touch it in light blue. All others are left white. The applet then gives you an approximate value for integral based on summing up average values of the function at the corners of the squares. You can use it to (a) check
you solutions to quiz and homework problems (b) develop ypur geometric insight,
and (c) to solve two variable integration problems numerically --
and understand limits in accuracy that one has with this method.
Applet that does curve plotting
in Cartesian coordinates This applet lets you specify up to three boundary functions in two variables. It then graphs the curve where each of these
equals zero in a different color -- so you can tell which curve is which --
and the region where all the functions are positive in yellow. The rest of the background is left white. In this case, the functions are specified in Cartesian coordinates, but the cursor reads of the coordinates of the pont it is at in
both Cartesian and polar coordinates. You can use it to check that you have set limits up right, or graphed a region correctly.
Applet that does curve plotting in polar coordinates This applet is the same as above, except that
the boundary functions are entered in polar form. You can use it to check that you have set limits up right, or graphed a region correctly.
About Pappus, last of the great Greek geometers
Third Section Notes
Section III.1: Finding Limits of Integration in Three Dimensions
in Cartesian Coordinates
Section III.2: Finding Limits of Integration in Three Dimensions
in Cylindrical and Spherical Coordinates
Fourth Section Notes
Section IV.1: Paths, Parameterized Curves,
and Parameterization
Section IV.2: Tangent and Normal Vectors of Curves in the Plane
Section IV.3: Vector Fields and
Line Integrals in Two Dimensions
Section IV.4: Work and Flux
in Two Dimensions
Section IV.5: More notes on
the Divergence and the Curl in Two Dimensions
Section IV.6: Equivalence of
the Divergence Theorem and Green's Theorem in the Plane
Related Materials
Applet on parameterization
and tangetial and normal components of vector fields This applet
allows you to enter a paramterized curve and a vector field. It then gives you a choice of viewing the tangent or normal components of the ambient vector field along the curve, the unit tangent and normal vector fields along the curve, etc.
Applet on
the realtive velocity approach to understanding the
divergecne and curl of vector fields This applet
allows you to enter a vector field, which is interpreted as a velocity field.
You can then click on a point, and it will graph the relative velocity field
for that point, as describrd in the section IV.6 notes.
About George Green
About Johann Carl Friedrich Gauss
Fifth Section Notes
Section V.1: Flux Density, Divergecne, and the Divergence Theorem in Three Dimensions
Section V.2: Circulation Density, Curl, and Stokes's Theorem in Three Dimensions
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