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Student Multivariate Calculus

Six new commands have been added and two previous commands have been augmented. The new commands are:

• 

CrossProduct

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diff

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DotProduct

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Norm

• 

Normalize

• 

TripleScalarProduct

 

The modified commands are MultiInt and SurfaceArea.

 

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The Norm and Normalize commands default to the Euclidean norm over the real numbers.

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The DotProduct and TripleScalarProduct commands calculate over the real numbers.

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These, and the CrossProduct command, support the Lines & Planes portion of the typical multivariate calculus course.

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Just as in the VectorCalculus packages, the top-level diff command now automatically maps over the components of a vector when implemented within this package.

 

The existing MultiInt and SurfaceArea commands have been updated with the capability to integrate over the same regions that are known to the modified int command in the VectorCalculus packages. The net effect of these improvements is to reduce the number of times a student of multivariate calculus needs to work outside the Student MultivariateCalculus package.

 

Maple can solve many problems in Multivariate Calculus using several methods: using typeset math, using the context sensitive menu, and lastly, by Maple commands. Solutions using each of these three methods are shown for the first three of the following examples.

 

Example 1: Computing a Norm and Normalizing a Vector

Example 2: Computing Cross Products, Dot Products and Triple Scalar Products

Example 3: Differentiating a Vector

Example 4: Integrating a Function Over a Region

Example 5: Calculating the Surface Area of a Region

Example 1: Computing a Norm and Normalizing a Vector

Step

Instructions

Results

Load the Student :-MultivariateCalculus package.

From the Tools menu, select Load Package, then Student Multivariate Calculus.

Loading Student:-MultivariateCalculus 

Define the vector V.

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Enter: V=a,b.

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Right-click on this expression and select Assign Name from the context menu.

V=a,bassign

Find the norm of V using typeset math.

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Type the norm bars.

• 

From the context menu, select Evaluate and Display Inline.

V = a2+b2

Alternatively, find the norm using context menu options.

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Type: V.

• 

From the context menu, select Evaluate and Display Inline.

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Again, from the context menu select Student Multivariate Calculus, then select Norm.

V = abnorma2+b2

Normalize V using the context menu.

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Type: V.

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From the context menu select Evaluate and Display Inline.

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Again, from the context menu select Student Multivariate Calculus then select Normalize.

 

V = abnormalizeaa2+b2ba2+b2

Use the Norm command to find the norm of V.

Type: "Norm(V)" and press Enter.

NormV = a2+b2

Use the Normalize command to normalize V.

Type: "Normalize(V)" and press Enter.

NormalizeV = aa2+b2ba2+b2

Example 2: Computing Cross Products, Dot Products and Triple Scalar Products

Step

Instructions

Results

Load the Student:-MultivariateCalculus package.

• 

From the Tools menu, select Load Package, then Student Multivariate Calculus.

Loading Student:-MultivariateCalculus.

Define Vectors A, B and C.

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Enter: A=a,b,c

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Right-click on this expression and select Assign Name from the context menu.

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Repeat for B and C (see Results column).

A=a,b,cassign

B=u,v,wassign

C=p,q,rassign

Vector Products via Typeset Math (See Common Symbols Palette).

Compute the vector products using typeset math.

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Context Menu: Evaluate and Display Inline.

Compute A·B using typeset math.

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Type: A.

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From the Common Symbols palette, select ·

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Type: B.

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Right-click on the expression.

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From the context menu select Evaluate and Display Inline.

A·B = au+bv+cw

Compute A×B using typeset math.

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Type: A.

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From the Common Symbols palette select ×

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Type: B.

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Right-click on the expression.

• 

From the context menu select Evaluate and Display Inline.

 

 

A×B = bwcvaw+cuavbu

Compute A·B×Cusing typeset math.

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Type: A·B×C, selecting the operators from the Common Symbols palette.

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Right-click on the expression.

• 

From the context menu select Evaluate and Display Inline.

 

 

A·B×C = aqw+rv+bpwru+cpv+qu

Vector Products via the Context Menu

Compute   A·B from the context menu.

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Enter: A,B.

• 

Right-click on A,B and select Evaluate and Display Inline from the context menu.

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Right-click on the output from the previous step  and select Student Multivariate Calculus, then
Dot Product.

 

A,B = abc,uvwdot productau+bv+cw

Compute A×B from the context menu.

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Enter: A,B.

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Right-click on A,B and select Evaluate and Display Inline from the context menu.

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Right-click on the output from the previous step and select Student Multivariate Calculus, then
Cross Product.

 

A,B = abc,uvwcross productbwcvaw+cuavbu

Compute A·B×C from the context menu.

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Enter: A,B,C.

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Right-click on A,B,C and select Evaluate and Display Inline from the context menu.

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Right-click on the output from the previous step  and select Student Multivariate Calculus, then
Triple Scalar Product.

 

A,B,C = abc,uvw,pqrscalar triple productaqw+rv+bpwru+cpv+qu

Vector Products via the DotProduct, CrossProduct, and TripleScalarProduct Commands

Compute   A·B using the DotProduct command.

 

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Type: DotProduct(A,B).

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Press Enter.

DotProductA,B = au+bv+cw

Compute A×B using the CrossProduct command.

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Type: CrossProduct(A,B).

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Press Enter.

CrossProductA,B = bwcvaw+cuavbu

Compute A·B×C using the TripleScalarProduct command.

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Type: TripleScalarProduct(A,B,C).

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Press Enter.

 

TripleScalarProductA,B,C = aqw+rv+bpwru+cpv+qu

Example 3: Differentiating a Vector

Step

Instructions

Results

Load the Student:-MultivariateCalculus package.

• 

From the Tools menu, select Load Package, then Student Multivariate Calculus.

Loading Student:-MultivariateCalculus 

Define vector V.

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Type: V=fx,gx 

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From the context menu, select Assign Name.

 

V=fx,gxassign

Define vector v.

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Type: v=xt,yt

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From the context menu select Assign Name

v=xt,ytassign

Differentiate via Typeset Math

Differentiate V.

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Type V.

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Select # from the Common Symbols palette.

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Press Enter.

V = ⅆⅆxfxⅆⅆxgx

Differentiate v.

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From the Calculus palette, select A..

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Replace A with v.

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Press Enter.

v. = ⅆⅆtxtⅆⅆtyt

Differentiate via Context Menu

Differentiate V.

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Enter: V

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Right-click on V.

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From the context menu, select Evaluate and Display Inline.

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Select the output, then from the context menu, select Student Multivariate Calculus, then Differentiate, then select With Respect To x.

V = fxgxdifferentiateⅆⅆxfxⅆⅆxgx

Differentiate v.

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Enter: v

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Right-click on v  . 

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From the context menu, select Evaluate and Display Inline.

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Select the output, then from the context menu, select Student Multivariate Calculus, then Differentiate, then select With Respect To x.

v = xtytdifferentiateⅆⅆtxtⅆⅆtyt

Differentiate via the Differentiation Operator in the Calculus Palette

Differentiate V. 

• 

From the Calculus palette, select the differentiation operator, ⅆⅆxf.

• 

Replace f with V.

• 

Right-click the expression and select Evaluate and Display Inline.

ⅆⅆ x V = ⅆⅆxfxⅆⅆxgx

Differentiate v.

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From the Calculus palette, select the differentiation operator, ⅆⅆxf.

• 

Replace f  with v and replace x with t.

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Right-click the expression and select Evaluate and Display Inline.

 

ⅆⅆ t v = ⅆⅆtxtⅆⅆtyt

Differentiation via the Modified diff Command

Differentiate  V. 

• 

Type "diff(V,x)".

• 

Press Enter.

diffV,x = ⅆⅆxfxⅆⅆxgx

Differentiate v.

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Type "diff(v,x)".

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Press Enter.

diffv,t = ⅆⅆtxtⅆⅆtyt

Example 4: Integrating a Function Over a Region

Integrate fx,y=x y over the upper half of the circle whose center is 1,2 and whose radius is R.

Instructions

Results

• 

From the Tools menu, select Load Package, then Student Multivariate Calculus.

Loading Student:-MultivariateCalculus 

Use the MultiInt Command.

MultiIntx y,x,y=SectorCircle1,2,R,r,θ,0, π,output=integral

∫0R∫0πrcosθ+1rsinθ+2rⅆθⅆr

MultiIntx y,x,y=SectorCircle1,2,R,r,θ,0, π,output=value = πR2+23R3

MultiIntx y,x,y=SectorCircle1,2,R,r,θ,0, π,output=steps

πR2+23R3

In addition to Circle and Sector, the MultiInt command "knows" the additional predefined regions Ellipse (to which Sector also applies), Parallelepiped (a rectangular box with sides parallel to the coordinate planes), Rectangle, Sphere, Tetrahedron, Triangle, and the general Region.

Example 5: Calculating the Surface Area of a Region

Calculate the surface area of that portion of the surface z=3x2/4y2/3 that is defined over the triangle whose vertices are 0,0,3,0,1,2.

Instructions

Result

• 

From the Tools menu, select Load Package, then Student Multivariate Calculus.

Loading Student:-MultivariateCalculus.

• 

Type: Z=3x2/4y2/3

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Right-click on Z and select Assign Name.

Z=3x2/4y2/3assign

Use SurfaceArea Command.

SurfaceAreaSurfacex,y,Z,x,y=Triangle0,0,3,0,1,2,output=plot,axes=frame,labels=x,y,z

SurfaceAreaSurfacex,y,Z,x,y=Triangle0,0,3,0,1,2,output=integral

∫01∫2x0169x2+16y2+36ⅆyⅆx∫13∫3x0169x2+16y2+36ⅆyⅆx

evalfSurfaceAreaSurfacex,y,Z,x,y=Triangle0,0,3,0,1,2 = 4.028651111