<Text-field style="Heading 1" layout="Heading 1">Initializations</Text-field>

Space curves: Plotting Vector-Valued Functions in Maple

Calc IV Lab Week One

Karen Donnelly

Saint Joseph's College

Load the plots library so that the animate command may be used.

Load VectorCalculus so that the vectors can be defined and then plotted with SpaceCurve.

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Define options for plotting and name opts to save typing in subsequent plotting commands:

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

<Text-field style="Heading 1" layout="Heading 1">How to Plot Vector Valued Functions (Curves in Space)with Maple</Text-field>

The easiest way to plot a curve in space in Maple is to use the SpaceCurve function which is part of the plots library.

For example, to plot the helix curve 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 for t between 0 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJSZwaTtGJy8lJ2l0YWxpY0dGPkY1RjVGK0Y1:

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

To See how the curve is traced out as t increases from 0 to 4*Pi, we can use the animate command applied to the SpaceCurve command. .

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print(); # input placeholder

Here we animate the above curve to show its orientation as it is drawn

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

A curve in the plane is just a special case a curve in space. If in the x-y plane, the k coordinate is 0. Thus the plot of 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 is given by

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

print(); # input placeholder

In animated form:

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

<Text-field style="Heading 1" layout="Heading 1">Exercise</Text-field>

This exercise is from Larson, Section 12.1

Graph the vector valued function r(t) = 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 over the range t = 0 to 4Pi. Then make a conjecture about each of the transformations of the graph without plotting the transformations!! Then, plot to check your answers. Correct your conjectures if you made a mistake.

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==

Part a) u(t) = 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.

Conjecture: The graph of the transformed curve will be the same, except the x-coordinateLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= (i component) will be shifted back by 2 .

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRIlBGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI1BhRidGL0YyRkE=

Part b) u(t) = 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.

Conjecture: The height of the helix will increase faster (twice as fast).

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUSNQYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YrLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Part c: u(t) = 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.

Conjecture: The orientation will be reversed and it will go downwards from the same starting point, instead of upwards.

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

Part d: u(t) = 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.

Conjecture: This helix will have center as the x axis instead of the z-axis.

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

Part e: u(t) = 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.

Conjecture : The helix will have a radius three times as wide as the original helix (radius 6 now instead of 2.)

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

<Text-field style="Heading 1" layout="Heading 1">Major Commands used</Text-field>

From VectorCalculus package

SpaceCurve

From plots package

animate

JSFH