Chapter 9: Vector Calculus
Section 9.7: Conservative and Solenoidal Fields
Example 9.7.10
If ux,y,z is a scalar potential for
F=2xz+y2 i+2 x y−z3 j+x2−3yz2 k
show that ∫CF·dr=uQ−uP, where C is given parametrically by the equations x=1+2 t, y=2 t2−t+3, z=2+t3, and P and Q are its endpoints when t=0,1, respectively.
Solution
Mathematical Solution
∫CF·dr
=∫0121+2 t2+t3+2 t2−t+32 dt
+∫0121+2 t2 t2−t+3−2+t33 dt
+∫011+2 t2−32 t2−t+32+t32 dt
=−∫0122t10−10t9+27t8+96t7−42t6+108t5+60t4−48t3+39t2+14t−28dt
=−20
Evaluating C at t=0 and t=1 provides the points P and Q, which are respectively 1,3,2 and 3,4,3.
Using the scalar potential ux,y,z=x2z+xy2−y z3 found in Example 9.7.8, calculate
uQ=32⋅3+3⋅42−4⋅33=−33 and uP=12⋅2+1⋅32−3⋅23= −13
so that uQ−uP=−33−−13=−20.
Maple Solution - Interactive
Using the scalar potential ux,y,z=x2z+xy2−y z3 found in Example 9.7.8, Table 9.7.10(a) evaluates the line integral ∫CF·dr and compares the value to uQ−uP, where P and Q are respectively the points 1,3,2 and 3,4,3.
Initialize
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Tools≻Tasks≻Browse: Calculus - Vector≻ Vector Algebra and Settings≻ Display Format for Vectors
Press the Access Settings button and select "Display as Column Vector"
Display Format for Vectors
Define the vector field F
Enter the components of F in a free vector. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To Vector Field
Context Panel: Assign to a Name≻F
2 x z+y2,2 x y−z3,x2−3 y z2 = →to Vector Field →assign to a nameF
Define the scalar potential as the function ux,y,z
Context Panel: Assign Function
ux,y,z=x2z+xy2−y z3→assign as functionu
Define the path C
Context Panel: Assign Name
X=1+2 t→assign
Y=2 t2−t+3→assign
Z=2+t3→assign
Form and evaluate ∫CF·dr
Write the name F and press the Enter key.
Context Panel: Student Vector Calculus≻Line Integral Complete the dialog as per
Context Panel: Student Vector Calculus≻Line Integral Complete the dialog as per the figure to the right.
Context Panel: Simplify≻Simplify
Context Panel: Evaluate Integral
F
→line integral
∫0141+2tt3+2+22t2−t+32+−t3+23+21+2t2t2−t+34t−1+3−32t2−t+3t3+22+1+2t2t2ⅆt
= simplify
−∫0122t10−10t9+27t8+96t7−42t6+108t5+60t4−48t3+39t2+14t−28ⅆt
=
−20
Calculate uQ−uP
Context Panel: Evaluate and Display Inline
u3,4,3−u1,3,2 = −20
Table 9.7.10(a) Verification that ∫CF·dr=uQ−uP
Maple Solution - Coded
Using the scalar potential ux,y,z=x2z+xy2−y z3 found in Example 9.7.8, Table 9.7.10(b) evaluates the line integral ∫CF·dr and compares the value to uQ−uP, where P and Q are respectively the points 1,3,2 and 3,4,3.
Install the Student VectorCalculus package.
withStudent:-VectorCalculus:
Set display of vectors via BasisFormat.
BasisFormatfalse:
Define F via the VectorField command.
F≔VectorField2 x z+y2,2 x y−z3,x2−3 y z2:
Use "arrow" definition of a function.
u≔x,y,z→x2z+xy2−y z3:
Sequence of names assigned sequence of values.
X,Y,Z≔1+2 t,2 t2−t+3,2+t3:
Use the LineInt and simplify commands to form and evaluate ∫CF·dr
simplifyLineIntF,PathX,Y,Z,t=0..1,output=integral
LineIntF,PathX,Y,Z,t=0..1 = −20
Use the eval command to obtain P and Q as lists, then remove the brackets by appending .
uevalX,Y,Z,t=1−uevalX,Y,Z,t=0 = −20
Table 9.7.10(b) Verification that ∫CF·dr=uQ−uP
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