Chapter 4: Partial Differentiation
Section 4.5: Gradient Vector
Example 4.5.2
Let w=x2+2 y2+3 z2 and let P be the point 1,1,1.
Obtain ∇w at P.
On the same set of axes, graph the level surface w=6 and ∇w at P.
At P, show that ∇w is orthogonal to the level surface w=6. Hint: Show that this gradient is orthogonal to the x- and y-coordinate curves through P.
Solution
Mathematical Solution
Part (a):
∇wx=a|f(x)P = wx(1,1,1)wy(1,1,1)wz(1,1,1) = 246
Part (b): See Figure 4.5.2.(a).
use plots, Student:-VectorCalculus in module() local V,p1,p2,p3; V:=RootedVector(root=[1,1,1],<2,4,6>); p1:=PlotVector(V,color=black); p2:=implicitplot3d(x^2+2*y^2+3*z^2=6,x=-3..3,y=-2..2,z=-2..2,style=surface); p3:=display(p1,p2,scaling=constrained,axes=frame,tickmarks=[6,8,8],orientation=[45,80,0]); print(p3); end module: end use:
Figure 4.5.2(a) Level surface and gradient
Part (c):
The coordinate curves
R1y=1y5−2 y2/3 and R2x=x14−x2/3
lie in the surface defined by w=6, pass through the point P, and project respectively to the grid lines x=1 and y=1, in the xy-plane.
At P on the surface, the vectors
T1=R1/1=01−2/3 and T2=R2/1=10−1/3
are tangent to the curves R1y and R2x, respectively. That the gradient vector is orthogonal to the coordinates curves R1y and R2x is shown by the vanishing of the dot products, that is, by the calculations
T1·∇wP=01−2/3·246 = 4−4=0 and T2·∇wP=10−1/3·246 = 2−2=0
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Context Panel: Assign name
w=x2+2 y2+3 z2→assign
Part (a)
Write w. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Differentiate≻Gradient (See Figure 4.5.2(b).)
Context Panel: Select Element≻1
Context Panel: Assign to a Name≻Gw
Figure 4.5.2(b) Gradient dialog
w = x2+2y2+3z2→gradient246→select entry 1246→assign to a nameGw
Part (b)
Type Gw and press the Enter key. Context Panel: Plots≻Arrow from point See dialog in Figure 4.5.2(c).
Figure 4.5.2(c) Dialog for graphing an arrow
Gw
246
→plot arrow
Type w=6 and press the Enter key. Context Panel: Plots≻Plot Builder Obtain a graph of the surface w=6.
Control-drag (or copy/paste) the arrow onto the graph of the surface.
w=6
x2+2y2+3z2=6
→
Part (c)
Write the equation w=6.
Context Panel: Evaluate and Display Inline
Context Panel: Solve≻Obtain Solutions for≻z
Context Panel: Assign to a Name≻Z
w=6 = x2+2y2+3z2=6 ⇒ →solutions for z13−3x2−6y2+18,−13−3x2−6y2+18→select entry 113−3x2−6y2+18→assign to a nameZ
Expression palette: Evaluation template
Context Panel: Assign to a Name≻R[1] or R[2], as appropriate.
1,y,Zx=a|f(x)x=1 = →assign to a nameR1
x,1,Zx=a|f(x)y=1 = →assign to a nameR2
Expression palette: Evaluation template Calculus palette: Differentiation operator
Context Panel: Simplify≻Simplify
Context Panel: Assign to a Name≻T[1] or T[2], as appropriate.
ⅆⅆ y R1x=a|f(x)y=1 = = simplify →assign to a nameT1
ⅆⅆ x R2x=a|f(x)x=1 = = simplify →assign to a nameT2
Common Symbols palette: Dot product operator Context Panel: Evaluate and Display Inline
T1·Gw = 0
T2·Gw = 0
Maple Solution - Coded
Install the Student MultivariateCalculus package.
withStudent:-MultivariateCalculus:
Define wx,y,z.
w≔x2+2 y2+3 z2:
Use the Gradient command to obtain ∇w and the eval command to obtain ∇wP.
Gw≔Gradientw,x,y,z=1,1,11
Use the RootedVector command to define V= ∇wP with its "tail" at P:1,1,1.
Use the PlotVector command to graph V.
Use the implicitplot3d command to graph the level surface w=−6.
Use the display command to merge the graphs of the surface and its gradient.
V≔Student:-VectorCalculus:-RootedVectorroot=1,1,1,Gw:p1≔Student:-VectorCalculus:-PlotVectorV,color=black:p2≔plots:-implicitplot3dw=6,x=−3..3,y=−2..2,z=−2..2,style=surface:plots:-displayp1,p2,scaling=constrained,axes=frame:
Obtain zx,y by applying the solve command to w=6.
Z≔solvew=6,z1:
Use the eval command to obtain z1,y in the position-vector form of the plane section x=1.
R1≔1,y,evalZ,x=1:
Use the eval command to obtain zx,1 in the position-vector form of the plane section y=1.
R2≔x,1,evalZ,y=1:
Use the simplify, eval, and diff commands to obtain the tangent vector T1=R1/1.
T1≔simplifyevaldiffR1,y,y=1:
Use the simplify, eval, and diff commands to obtain the tangent vector T2=R2/1.
T2≔simplifyevaldiffR2,x,x=1:
Use the DotProduct command to show T1⊥∇wP.
DotProductT1,Gw = 0
Use the DotProduct command to show T2⊥∇wP.
DotProductT2,Gw = 0
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