rlopez

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Dr. Robert J. Lopez, Emeritus Professor of Mathematics at the Rose-Hulman Institute of Technology in Terre Haute, Indiana, USA, is an award winning educator in mathematics and is the author of several books including Advanced Engineering Mathematics (Addison-Wesley 2001). For over two decades, Dr. Lopez has also been a visionary figure in the introduction of Maplesoft technology into undergraduate education. Dr. Lopez earned his Ph.D. in mathematics from Purdue University, his MS from the University of Missouri - Rolla, and his BA from Marist College. He has held academic appointments at Rose-Hulman (1985-2003), Memorial University of Newfoundland (1973-1985), and the University of Nebraska - Lincoln (1970-1973). His publication and research history includes manuscripts and papers in a variety of pure and applied mathematics topics. He has received numerous awards for outstanding scholarship and teaching.

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These are answers submitted by rlopez

The following commands will do what has been asked.

with(Student:-VectorCalculus):
f:=4*x^2+9*y^2:
V:=evalVF(Gradient(f),<2,1>):
p1:=PlotVector(V):
p2:=plots:-implicitplot(f=25,x=-4..4,y=-2..2):
plots:-display(p1,p2,scaling=constrained)
 

The Gradient command in the Student VC package produces a VectorField. The correct way to evaluate a VectorField at a point is to use the evalVF command, which produces a RootedVector, one that remembers where its "tail" is. The easiest way to graph a vector is to use the PlotVector command. Of course, there's a Gradient command in the Student MultivariateCalculus command, but it would create a vector whose "tail" is at the origin, and this would have to be followed by an arrow command in either the plots package or the plottools package. (Both these packages have an arrow command, but each uses different syntax that I can never remember. I always use PlotVector because of its simplicity.)

If I read the worksheet correctly, there are three equations in P, V, W, and t; ln(V) and ln(W) both appear in the third equation. I suppose that an invocation of the implicit function theorem will show that a solution for P(t), V(t), W(t) exists, but I doubt that it can be found explicitly. It's not that Maple can't do it, but rather, that there is no set of algebraic manipulations that can do it.

If graphs of P(t),V(t), W(t) are all that are needed, then look into applying the Draghilev method. Search for it in MaplePrimes, or see the Reporter article here that's stored in the collection of MapleApps (www.MapleApps.com). The article is easily found by going to MapleApps and doing a simple search on "Dragilev".

(Unfortunately, the article dropped the "h" in the name "Draghilev," an error that was subsequently pointed out but not corrected in MapleApps.)

 

The worksheet at the end of the link below shows how to use the syntax-free Direction Field Task Template. From the earlier responses, it seemed to me that the issue in using the DEplot command was mastery of the syntax. The built-in Task Template is a way to avoid the struggle with syntax.

Direction_Field_Task_Template.mw

The worksheet in the link below contains an example I just happened to have completed just before this present thread appeared. It's complete, but not "pretty."

In short, the line integral of a scalar geometrically is the area of a surface defined by the path of integration and the graph of the function f(x,y) that is the integrand. The example in the worksheet shows that indeed, this is true. And if you write the Riemann sum for the integral and visualize the rectangles in the surface whose area is purportedly being computed, you can easily accept that this line integral does measure surface area.

Line_Int_of_Scalar-meaning.mw.

Try something along the following lines.

with(Student:-VectorCalculus);
C := PositionVector([t, 0, (2*t^(3/2))/3]):
V := PlotPositionVector(C, 0 .. 2, tangent = true, points = [1], tangentoptions = [color = green]):
plots:-display(V, view = [0 .. 2, -1 .. 1, 0 .. 2], labels = [x, y, z]);

The help page for PlotPositionVector points out that arrows of various fields along the curve can be graphed. Moreover, the arrows of arbitrary VectorFields can be drawn. This is a very versitile command that has built into it features of the primitives such as spacecurve and arrow. Note also that this package contains the SpaceCurve command that draws what spacecurve in the plots package draws, but also draws a plane curve with the same syntax, something that spacecurve does not do.

https://www.bing.com/videos/search?q=lines+and+planes+Maplesoft+Youtube&docid=607991258261816461&mid=5BE57C2C787E7B8ECE5E5BE57C2C787E7B8ECE5E&view=detail&FORM=VIRE

Click here

At 52 minutes, 25 seconds into this video, Problem 10 (distance from a point to a plane) is solved via the tools in the Student MultivariateCalculus package. All 11 problems in this video illustrate a variety of tools in this package related to the typical section in a calculus text "Vectors, Lines, and Planes."

with(Student:-VectorCalculus):

p1:=<4,3,-5>:
p2:=<3,-1,4>:
v:=p2-p1:
V:=RootedVector(root=p1,v):
PlotVector([p1, p2, v, V], color = [black, red, green, gold], width = 0.2, scaling = constrained, labels = [x, y, z])

The vector V retains as an attribute the location of its initial point. The parallel transport of v to its new location is thereby built into the definition of V, and is not an artifact of how it is graphed. 

Student:-MultivariateCalculus:-LagrangeMultipliers((x-2*y)/(5*x^2-2*x*y+2*y^2), [2*x^2 - y^2 + x*y-1], [x,y])

Include the option "output=detailed" to get the values of the function

and the multipliers at the critical points.

Alternatively, 

F:=(x-2*y)/(5*x^2-2*x*y+2*y^2)+lambda*([2*x^2-y^2+x*y-1)

solve({diff(F,x)=0, diff(F,y)=0, diff(F,lambda)=0},{x,y,lambda})

I'd use the following approach.


with(Student:-VectorCalculus):
r := <cos(t), sin(t), t>;
rp := diff(r, t);
SpaceCurve(r, t = 0 .. 2*Pi);
SpaceCurve(rp, t = 0 .. 2*Pi);

 

No need to make r a function just to draw a graph. The SpaceCurve command in VectorCalculus is more robust than the spacecurve command in plots since it applies top both 2- and 3-component vectors.

For any task template that uses embedded components, the code is "behind the components." By this is meant, right-click on the component or use the Context Panel, and select the option that contains the code.

Most Math Apps are written with embedded components, but are coded with procedures defined in a start-up section, with simple function calls "behind the components." Unfortunately, it does not seem possible to access this start-up code, but some other contributor to this forum might know a trick or two that I don't.

Tutors have a command-name in the associated Student packages. Showstat can be applied to that command name.

An Assistant such as the ODE Analyzer Assistant has the command name dsolve[':-interactive'], to which the showstat command can be applied. These command names can be found as follows.

At a (red) worksheet prompt, using 1D math input, enter a differential equation and use the Context Panel to select the interactive form of solving the DE. Maple will write the command that calls the ODE Analyzer Assistant. This name can also be found by rummaging through the appropriate help page, but having the Context Panel print the name might be faster.

answer5 contains equations x=..., y=...

Your input to arctan is then the ratio of two equations, not of the two numbers on the right of the equal signs.

A simple fix would be eval(arctan(y/x),answer5)

The following lines produce the result you want.

with(Typesetting):
Settings(useprime, prime=x,typesetprime=true):
Suppress(f(x)):
D(f)(x)(0)

 

The VectorCalculus package requires explicit declaration of the coordinate system and coordinate names. These names are used, for example, when Maple then computes a gradient.

The Student VectorCalculus package defines default variable names in the Cartesian, polar, cylindrical and spherical coordinate systems. Hence, there is far less need for declaring coordinate systems and their associated variable names. (In Cartesian coordinates, the default coordinate names are x, y, z.)

So, what you tried in VectorCalculus would have worked in the Student package, without the need to include differentiation variables in the call to gradient of f. The default names x and y would have been used in both instances of the nabla.

The error you are  making is not understanding the help page for pdsolve/numeric, wherein the very first line is

PDEsys-single or set or list of time-dependent partial differential equations in two independent variables.

This line means that only evolution equations (i.e., heat and wave equations) can be solved numerically. There is no numeric elliptic solver in Maple.
 

From the Maple Application Center, find Prof Werner's Fourier Series package:

Symbolic Computation of Fourier Series

It contains a command for generating and animating the graph of Fourier series. I highly recommend this package and had always hoped that something like it would be added to Maple.

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