## 2670 Reputation

16 years, 165 days

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.

## Graphing vectors with VectorCalculus pac...

The simplest way to graph vectors is to use the appropriate commands in the VectorCalculus package. I prefer the Student VectorCalculus package because it is more forgiving with respect to the need for defining coordinate systems and coordinate-variable names.

The commands to use are PlotVector and RootedVector. The PlotVector command will graph various kinds of vectors, including ones defined simply with angle brackets: <1,2,3>. The RootedVector command will attach a starting point to the definition of the vector so that you don't have to make the adjustments needed when using either of the "arrow" commands in Maple, one in the plottools package, and one in the plots package.

In addition to these two commands, I recommend the PositionVector and PlotPositionVector commands. The PositionVector command is for defining a curve or surface as a position vector. The PlotPositionVector command then draws the curve or surface, and admits the addition of arrows from a variety of vector fields defined along the curve or surface. These two commands are an extremely powerful and useful pair of visualization tools.

I have just completed an extensive project on surface curvature in which I installed the Student MultivariateCalculus package to get access to a CrossProduct and DotProduct command. I also used the alias command to define a shortcut to the commands I wanted in VectorCalculus. This way, I avoided the conflict between the simple dot and crossproduct operations in the MultivariateCalculus package, and the more entangled versions in VectorCalculus. So, I'd make calculations along the following lines.

with(Student:-MultivariateCalculus):
alias(VC=Student:-VectorCalculus):

A:=<1,2,3>:
B:=VC:-RootedVector(root=[1,2,3],<-2,3,1>):

VC:-PlotVector([A,B],color=[black,red],width=.1)

## asympt...

Took the function y=sqrt(1+x^2) (suggested by hyperbola, which has an asymptote), applied the asympt command to get its expansion about the point at infinity, and took the limit of the ratio of y to the first term in the asymptotic expansion, and got 1.

y:=sqrt(1+x^2);
asympt(y,x,2) # the "2" determines the order of the expansion. First term is x

limit(y/x,x=infinity)=1

## Control= or Context Panel...

In Windows, Control= will implement "Evaluate and Display Inline", and this option also appears in the Context Panel.

## Jacobian...

There are several errors and misconceptions in the post.

The with command should not be followed by a multiplication operator. It would be entered as with(VectorCalculus), for example.

There is no package named Linalg. There is a deprecated package "linalg" but please don't use that package - it's no longer supported.

There is a Jacobian command in both the VectorCalculus, and the Student:-MultivariateCalculus packages. But they have different syntax, so read the help pages.

There is no Jacobian command in the LinearAlgebra package, although there was a "jacobian" command in the old linalg package.

## definitions of curvature...

The page "Definition,curvature" is from a third-party work that Maple incorporated into its help system many years ago. There is no guarantee that the definitions the Maple developers use when coding procedures match the definitions in that third-party work.

That being said, newer calculus books typically define curvature with |y"| in the numerator, so the curvature is always positive. This is in keeping with the generalization to higher dimensions where curvature of a space curve is defined as the length (necessarily positive) of a "curvature" vector.

I had an old calculus text (1927?) that defined plane curvature with just y" in the numerator, so there has been a drift in what is taken as "correct" over the years.

## DynamicSystems:-DiscretePlot...

The DiscretePlot command in the DynamicSystems package will draw the graph. This was detailed in a Maple Reporter article in November of 2011. Here's a link to the article that is stored in MapleApps.

https://www.maplesoft.com/applications/view.aspx?SID=127613&ref=Feed

## Avoid use of the old student package...

Again, please avoid using the old student package. It hasn't been actively supported for more than 15 years.

The following command will immediately evaluate to the correct answer.

int(1/(1+x+y+z)^3,[z=0..1-x-y, y=0..1-x, x=0..1])

Also relevant are the following two task templates:

## Significant Maple (and mathematical) mis...

Please stop using the unsuppprted and outdated student package. It's not been maintained for more than 15 years.

Maple does not recognize "inf" as the symbol for "infinity." Type out the full word or, in 2D math, use the appropriate symbol from the Common Symbols palette.

If you want to integrate over the first quadrant portion of the interior of a circle of radius "a", do not then integrate over the whole of the first quadrant.

The following command will immediately return the value of the integral.

int(1/sqrt(a^2-x^2-y^2),[y=0..sqrt(a^2-x^2),x=0..a])

There are two task-templates that are of interest here.

## Turn help page into a worksheet...

If you turn the help page into a worksheet (click the icon in the toolbar), you then can make changes in style. Unfortunately, you have to do this for each help page as you bring it up.

## infnorm in the numapprox package...

I believe the built-in command you want to use is "infnorm" in the numapprox package.

I tried it on the test example in this thread and it produces the correct result.

## Spherical coordinate variables are posit...

The thing to keep in mind when using spherical coordinates in the VectorCalculus packages is that the names of the variables are irrelevant. It's their position in the triple spherical[u,v,w] that determines the definitions of the variables. The first name in the triple is the radial variable; the second, the angle down from the z-axis; and the third, the angle around the z-azis.

Calling the middle variable "theta" as you have done with the command SetCoordinates('spherical'[r, theta, phi]), changes the name of the "dropping down" angle to theta, but doesn't make theta match the very first image in your post.

One of the benefits of using the Student VectorCalculus package is that for Cartesian, polar, cylindrical, and spherical systems, there are default coordinate variable names recognized. For spherical, it's [r, phi, theta], with phi, being the middle name, the angle down from the z-axis, as in the diagram in your post. Your figure follows the definition of spherical coordinates found in most math books. Physics and engineering texts tend to interchange the names of the two angles. Thus, they make theta the angle down from the z-axis, and phi the angle around the z-axis.

So, the engineer of physicist using the VectorCalculus package would define a vector field in spherical coordinates with the syntax

VectorVield(<f(r,theta,phi),g(r,theta,phi),h(r,theta,phi)>,coords=[r,theta,phi])

This overwrites the defaults in the Student package and precludes the need for SetCoordinates in the VectorCalculus package.

## Default symbol for imaginary unit...

The default symbol for the imaginary unit (sqrt(-1)) is the letter I, in upper case. In the Common Symbols palette you see the three characters i, j, and I, all of which point to the default imaginary unit name, I. If you enter either the i or the j and hit the Enter key, the echo will be I.

So, in your trial worksheet, you typed the letter i, which is not interpreted as the i in the Common Symbols palette. It's just the letter i.

If the interface command is used to change from I to i as sqrt(-1), then a typed i would point to sqrt(-1).

In other words, what the symbols in the paletts do is establish a "dictionary" in which the palette symbol points to some underlying thing in Maple. If you were to type many of those same symbols, the typed character would not necessarily point to the appropriate underlying Maple item.

Finally, try hovering over the i, j, and I in the Common Symbols palette. You should see that all three point to the same thing, namely, the default imaginary unit, I.

## Use Student:-MultivariateCalculus tools...

with(Student:-MultivariateCalculus):

L:=Line([x1,y1,z1],[x2,y2,z2]);

simplify(Projection([x3,y3,z3],L));

In fact, these two calculations can be done in a syntax-free way via the Context Panel once the Student:-Multivariate package is loaded (which can also be done syntax-free via the Tools/Load Package menu).

Of course, the detailed solutions provided by the other contributors give insight into the math that the solution by built-in functions perhaps hides. But I always find it helpful to obtain an answer first, with a minimum of "fuss" before I dig in to see how the math works.

## PlotPositionVector...

In the VectorCalculus package (Student or otherwise) there are two relevant commands, namely, PositionVector, and PlotPositionVector. The first is used to define a curve (or surface) parametrically; and the second, to graph the curve (or surface) along with any number of different vector fields. Here, the vector field to include would be tangents. Of course, this still requires two commands, and a look at the help pages for getting the syntax right, but I find this pair of commands to be both powerful and useful.

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