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d mv , where m is the mass of an object and v is its velocity. dt When F has the representation F r , it is said to be conservative. The previous theorems tell us that the integrals of conservative elds of force are independent of path. Furthermore, showing that r F 0 is the preferred way of showing that F is conservative, since it involves di erentiation, while demonstrating that exists such that F r requires integration. EXAMPLE. Newton s second law for forces is F

g1 = Plot[1 + Sin[x] {x, 0, 2 o}]; , g2 = Plot[2 + Sin[x] {x, 2 o, 4 o}]; , g3 = Plot[3 + Sin[x] {x, 4 o, 6 o}]; , Show[g1, g2, g3, PlotRange Automatic]

Standard geometric shapes can be constructed with the Graphics command and viewed with the Show command.

The following are a few of the more common graphics primitives available in Mathematica:

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SURFACE INTEGRALS Our previous double integrals have been related to a very special surface, the plane. Now we consider other surfaces, yet, the approach is quite similar. Surfaces can be viewed intrinsically, i.e., as non-Euclidean spaces; however, we do not do that. Rather, the surface is thought of as embedded in a three-dimensional Euclidean space and expressed through a two-parameter vector representation: x r v1 ; v2 While the purpose of the vector representation is to be general (that is, interpretable through any allowable three-space coordinate system), it is convenient to initially think in terms of rectangular Cartesian coordinates; therefore, assume r xi yj zk and that there is a parametric representation x r v1 ; v2 ; y r v1 ; v2 ; z r v1 ; v2 11

Circle[{x, y}, r] creates a circle centered at (x, y) having radius r. Disk[{x, y}, r] creates a disk (filled circle) centered at (x, y) having radius r. Point[{x, y}] plots a point at coordinate (x, y). Line[{{x1, y1}, {x2, y2}, ...}] draws lines connecting points (x1, y1), (x2, y2), . . . Rectangle[{x1, y1}, {x2, y2}] creates a filled rectangle having (x1, y1) and (x2, y2) as opposite ends of a diagonal. Polygon[{{x1, y1}, {x2, y2}, ...}]constructs a filled polygon having points (x1, y1), (x2, y2), . . . as vertices. Text[textstring, {x, y}] prints a string of text centered at position (x, y). TextStyle allows you to change the default font and size used in the graph s text. TextStyle {FontFamily fontname, FontSize size} is a simple, but useful, application.

1. The first five alphanumeric characters of the UNIX file name are uppercased and used as the root of the new file name. Non-alphanumeric characters are ignored. 2. A "~" or the character designated by mangling char parameter is appended to the new five-character root file name and is followed with a two-character hash of the original file name without any extension. 3. The first three characters of the original file extension are uppercased and used as the new file extension. If there are no periods in the original file name, then no extension is applied. 4. If not specified as hidden, file names with a leading period (UNIX hidden files) are treated as above but have the leading period removed and have an extension of three "_" characters.

When viewing graphics objects using Show, the default, Axes False, causes the object to be drawn without axes. If desired, Axes True may be included as an option.

g1 = Graphics[Circle[{0, 0}, 1]]; g2 = Graphics[Line[{{ 1, 1}, { 1, 1}, {1, 1}, {1, 1}, { 1, 1}}]]; g3 = Graphics[Polygon[{{ 1, 0}, {0, 1}, {1, 0}, {0, 1}}]];

The functions are assumed to be continuously di erentiable. The parameter curves v2 const and v1 const establish a coordinate system on the surface (just as y const, and x const form such a system in the plane). The key to establishing the surface integral of a function is the di erential element of surface area. (For the plane that element is dA dx; dy.) At any point, P, of the surface dx @r @r dv dv @v1 1 @v2 2

g4 = Graphics[Text["Square in a Circle in a Square", {0, 1.2}, TextStyle {FontSize 20}]];

Show[g1, g2, g3, g4]

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