Visualizing Directional Derivatives and Gradients
\302\251Mike May, S.J., Saint Louis University, 2006 - maymk@slu.edu
As we have been looking at directional derivatives and gradients, it seems worthwhile to look at a Maple visualization of everything we have been doing.
First we want to load the appropriate commands. We will need commands from the package plots. (Restart cleans up in case we were already running a Maple worksheet.)
restart: with(plots):
Directional derivatives:
For preliminaries we want to define the function, the point in the x-y plane we are evaluating at and the direction vector we are interested in taking a derivative in.
f := (x,y) -> x^2 + 2*y^2;
xval :=2: yval:= 3:
dirvec := <3, 4>;
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Next we would like to compute a unit vector in the desired direction and the z-coordinate of the point of interest.
uvec := dirvec/sqrt(dirvec.dirvec);
zval := f(xval, yval);
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IiNB
To visualize what we are doing, we look at the surface together with a plane containing the direction vector.
surf:= plot3d(f(x,y), x=xval-2..xval+2,
y=yval-2..yval+2,
axes=boxed, color=red, style = patchnogrid):
pathc :=spacecurve(
[xval+t*uvec[1], yval+t*uvec[2],
f(xval+t*uvec[1], yval+t*uvec[2])],
t=-2..2, color=blue, thickness=2):
cutplane := plot3d([xval+u*uvec[1],
yval+u*uvec[2],zval+v],
u=-2..2,v=-5..5, color=yellow, style=
patchnogrid):
display3d({cutplane, surf, pathc});
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The surface is in red, the plane that cuts the cross section in the desired direction is in yellow, and the curve in the cross section is in blue.
We can get a better view if we look at the function as a function of one variable in the cutting plane.
plot(f(xval+t*uvec[1],yval+t*uvec[2]), t=-2..2);
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
We can compute the directional derivative from the definition.
dderiv := limit((f(xval+t*uvec[1], yval+t*uvec[2]) - f(xval, yval))/t, t=0);
IiM3
This can be used to plot a tangent line to the curve.
plot({f(xval+t*uvec[1],yval+t*uvec[2]), zval+dderiv*t}, t=-2..2);
NiktJSdDVVJWRVNHNiQ3UzckJCEjPyEiIiQiJGMlISIjNyQkITJMTExMUTZHIj4hIzskIjE2UG07SnhZXSEjOjckJCEybW1tVC5ccCQ9ISM7JCIxJFJtbm0oZiFcJiEjOjckJCEyTExMJCkpUWpePCEjOyQiMThGO1BpRjdnISM6NyQkITJLTEwkPUt2bDshIzskIjEnMydSJ3lDOmMnISM6NyQkITJvbW1UcyFHIWUiISM7JCIxbks7a0E9S3IhIzo3JCQhMk1MTDN5TzVdIiEjOyQiMXQoZlhOaEVvKCEjOjckJCEsdkUlKSo9OSEjNSQiMT5sSjFjTnUjKSEjOjckJCEyTUxMM1dEVEwiISM7JCIxRTZvV2FeNCopISM6NyQkISx2dlEmXDchIzUkIjE8N2xrVDltJiohIzo3JCQhMm1tbW0mNGBpNiEjOyQiMWtUJDRHMG0tIiEjOTckJCEyTExMJFFXKmUzIiEjOyQiMmFMPHosNS40IiEjOjckJCExLCsrKygpPicqKiohIzskIjJkIXkwKFxKVjsiISM6NyQkISkwIipIIiohIikkIjFQLiJlUDg2QyIhIzk3JCQhKTgzJkgpISIpJCIyKT4oKSlwKGZWPDghIzo3JCQhMUxMTDNrKHBgKCEjOyQiMUdHJUhwQygpUSIhIzk3JCQhMW5tbW1qXk5tISM7JCIxdTwtanQlZloiISM5NyQkITFvbW1tOSc9KGUhIzskIjIoM3RoJnlAPmIiISM6NyQkITEsKyt2I1xOKVwhIzskIjE+cyQqUVpxVTshIzk3JCQhMidvbW1tQ0MoPiUhIzwkIjJPTSJwR0RBRDwhIzo3JCQhMSoqKioqXEZSWEwkISM7JCIyKSlbcGxwITQ9PSEjOjckJCErRD0vOEQhIzUkIjJvRzZIPyN6Mz4hIzo3JCQhMmltbW1UJiplbCIhIzwkIjIjUVoqSFQqeTA/ISM6NyQkITFwbW07V24obykhIzwkIjFHSUUzZClwNCMhIzk3JCQhMi5QTExMZVYoPiEjPiQiMScqb0c0OWooPiMhIzk3JCQiMU1tbTtmYEAnKSEjPCQiMkVpZHVYeFlJIyEjOjckJCIxKSoqKipcblopSDshIzskIjJBblpFQVEqKlIjISM6NyQkIjJZbW1tSnkqZUMhIzwkIjJDVGAqelAqXF0jISM6NyQkIjInKSoqKioqKlJeYkokISM8JCIxNDYpSFwlKmVoIyEjOTckJCIxJyoqKioqXDVhYFQhIzskIjE8bUxaIT1ucyMhIzk3JCQiMSgqKioqXDdSVidcISM7JCIyKEdaQlV6OE9HISM6NyQkIjEnKioqKipcQGZrZSEjOyQiMHkhUSxqOmdIISM4NyQkIjBMTExgNE5uJyEjOiQiMVhnT1MoZlEyJCEjOTckJCIxKioqKioqKlwsc2AoISM7JCIyeGkhZTE7aig+JCEjOjckJCIxbG1tInpNKT4kKSEjOyQiMUdhZC0vIT5KJCEjOTckJCIxKioqKioqKnBmYTwqISM7JCIyOVBlTkFEIlJNISM6NyQkIjFITExlZ2AhKSoqISM7JCIyTjw+NWBFNWMkISM6NyQkIjIpKioqKlwjRzJBMyIhIzskIjFcbHhYNXMhcCQhIzk3JCQiMktMTEwpR1trNiEjOyQiMTpYQi5vdz5RISM5NyQkIjIpKioqKlw3eWhdNyEjOyQiMU5Ec1NaQ2RSISM5NyQkIjJvbW1tKWZkTDghIzskIjEiNC4ncEcmPjQlISM5NyQkIjJtbW07cTclPTkhIzskIjEiKilHXiZmL0tVISM5NyQkIjJMTExlI3BhLTohIzskIjJtSl1CWTVMUCUhIzo3JCQiKmFkKXo6ISIpJCIxY3omRyNbOzBYISM5NyQkIjJMTEwkR1VZbzshIzskIjEjUUBsI2VwZVkhIzk3JCQiMmxtbW01Onh1IiEjOyQiMk5pRmFYKD4peiUhIzo3JCQiLEQyOEEkPSEjNSQiMVQmZVtZLyNcXCEjOTckJCIsdlMpMzg+ISM1JCIyWVEwKXkzJGY0JiEjOjckJCIjPyEiIiQiMiYqKioqKioqKioqKmZEJiEjOi0lJkNPTE9SRzYmJSRSR0JHJCIjNSEiIiQiIiEhIiIkIiIhISIiLSUnQ1VSVkVTRzYkN1M3JCQhIz8hIiIkISM/ISIiNyQkITJMTExMUTZHIj4hIzskITEjKioqKioqKmZPUCYqISM7NyQkITJtbW1ULlxwJD0hIzskITAnKSoqKioqNCUpUVYhIzs3JCQhMkxMTCQpKVFqXjwhIzskIjEiKioqKioqUkxSISkqISM7NyQkITJLTEwkPUt2bDshIzskIjI3KysrUWg0LCMhIzs3JCQhMm9tbVRzIUchZSIhIzskIjEoKioqKio0OGpPSSEjOjckJCEyTUxMM3lPNV0iISM7JCIyKSkqKioqKkgnZXYpUiEjOzckJCEsdkUlKSo9OSEjNSQiMSwrKyF6KT1zXCEjOjckJCEyTUxMM1dEVEwiISM7JCIqciVcISpmISIpNyQkISx2dlEmXDchIzUkIjEpKioqKio0XGAwcSEjOjckJCEybW1tbSY0YGk2ISM7JCIqX0cnXCEpISIpNyQkITJMTEwkUVcqZTMiISM7JCIqdW0jcCopISIpNyQkITEsKysrKCk+JyoqKiEjOyQiMioqKioqKmY6Yy8rIiEjOjckJCEpMCIqSCIqISIpJCIqdTVXNSIhIig3JCQhKTgzJkgpISIpJCIyKioqKioqUkMhZi83ISM6NyQkITFMTEwzayhwYCghIzskIitKR2MmSCIhIik3JCQhMW5tbW1qXk5tISM7JCIrTyFRUFMiISIpNyQkITFvbW1tOSc9KGUhIzskIjIpKioqKipSaXdgXCIhIzo3JCQhMSwrK3YjXE4pXCEjOyQiKygzdT5nIiEiKTckJCEyJ29tbW1DQyg+JSEjPCQiKy80TCdwIiEiKTckJCExKioqKipcRlJYTCQhIzskIisoR2IpKnoiISIpNyQkIStEPS84RCEjNSQiKyIpXFYpKj0hIik3JCQhMmltbW1UJiplbCIhIzwkIipiI0gsPyEiKDckJCExcG1tO1duKG8pISM8JCIyKSoqKioqcCF6dSY0IyEjOjckJCEyLlBMTExlVig+ISM+JCIqeEl3PiMhIig3JCQiMU1tbTtmYEAnKSEjPCQiMicqKioqKjRWZU1JIyEjOjckJCIxKSoqKipcblopSDshIzskIitAPGUmUiMhIik3JCQiMlltbW1KeSplQyEjPCQiMigqKioqKnpSeF1cIyEjOjckJCIyJykqKioqKipSXmJKJCEjPCQiK29oJ3lmIyEiKTckJCIxJyoqKioqXDVhYFQhIzskIjImKioqKipmI1xVKXAjISM6NyQkIjEoKioqKlw3UlYnXCEjOyQiMicqKioqKlxwP2R6IyEjOjckJCIxJyoqKioqXEBma2UhIzskIjInKioqKip6MF5QIUghIzo3JCQiMExMTGA0Tm4nISM6JCIyKCoqKioqUjlAMyskISM6NyQkIjEqKioqKioqXCxzYCghIzskIio9a1c1JCEiKDckJCIxbG1tInpNKT4kKSEjOyQiMikqKioqKlw8IVEpPiQhIzo3JCQiMSoqKioqKipwZmE8KiEjOyQiK2teMCxMISIpNyQkIjFITExlZ2AhKSoqISM7JCIrRlZtKFIkISIpNyQkIjIpKioqKlwjRzJBMyIhIzskIjEqKioqKipRKFsnKVwkISM5NyQkIjJLTExMKUdbazYhIzskIipZenRmJCEiKDckJCIyKSoqKipcN3loXTchIzskIit2OHUrUCEiKTckJCIyb21tbSlmZEw4ISM7JCIrJT0iSCtRISIpNyQkIjJtbW07cTclPTkhIzskIitVXzQtUiEiKTckJCIyTExMZSNwYS06ISM7JCIrNmowLlMhIik3JCQiKmFkKXo6ISIpJCIrWyFIZTQlISIpNyQkIjJMTEwkR1VZbzshIzskIit1cTotVSEiKTckJCIybG1tbTU6eHUiISM7JCIrRyJlc0glISIpNyQkIixEMjhBJD0hIzUkIisob2InKVIlISIpNyQkIix2UykzOD4hIzUkIisqMzFkXCUhIik3JCQiIz8hIiIkIiRnJSEiIi0lJkNPTE9SRzYmJSRSR0JHJCIyJ1xfJ2VNJXlnPiEjPCQiMWlqOS5lQFIhKSEjOyQiMidcXydlTSV5Zz4hIzwtJSVWSUVXRzYkOyQhIz8hIiIkIiM/ISIiJShERUZBVUxURy0lK0FYRVNMQUJFTFNHNidJInRHNiJRITYiLSUlRk9OVEc2JSUhRyUhRyIjNSUrSE9SSVpPTlRBTEclK0hPUklaT05UQUxHLSUqQVhFU1NUWUxFRzYjJSdOT1JNQUxHLSUoU0NBTElOR0c2IyUuVU5DT05TVFJBSU5FREctJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiRTIiEiIi0lKUJPVU5EU19ZRzYjJCIjISohIiItJS1CT1VORFNfV0lEVEhHNiMkIiVxUCEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiU/TyEiIi0lKUNISUxEUkVORzYi
We can also put this line back on the 3D graph of the surface.
tanline :=spacecurve([xval+t*uvec[1],
yval+t*uvec[2], zval+t*dderiv],
t=-2..2, color=green, thickness=3):
display3d({cutplane, surf, pathc, tanline});
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For convenience sake we can put everything together into one block of code.
# change first three lines in cut and paste
# mode for exercises
f := (x,y) -> x^2 + 2*y^3;
xval :=2: yval:= 1:
dirvec := <3, -1>:
# this is a comment to indicate the first
# three lines can be changed by the
# student.
uvec := dirvec/sqrt(dirvec.dirvec);
zval := f(xval,yval);
dderiv := limit((f(xval+t*uvec[1], yval+t*uvec[2]) - f(xval, yval))/t, t=0);
plot([f(xval+t*uvec[1],yval+t*uvec[2]), zval+dderiv*t],t=-2..2,
legend=["f restricted to u", "directional derivative"]);
surf:= plot3d(f(x,y), x=xval-2..xval+2, y=yval-2..yval+2, axes=boxed,
color=red, style = patchnogrid):
pathc := spacecurve(
[xval+t*uvec[1], yval+t*uvec[2], f(xval+t*uvec[1], yval+t*uvec[2])],
t=-2..2, color=blue, thickness=2):
cutplane := plot3d([xval+u*uvec[1], yval+u*uvec[2],zval+v],
u=-2..2,v=-5..5, color=yellow, style=patchnogrid):
tanline :=spacecurve([xval+t*uvec[1], yval+t*uvec[2], zval+t*dderiv],
t=-2..2, color=green, thickness=3):
display3d({cutplane, surf, pathc, tanline});
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Gradients:
The gradient of a function can be thought of in a number of ways. The easiest description is symbolic. The gradient of f(x,y) is defined to be the vector with components equal to the partial derivatives. In the terminology we have been using, grad(f) = [x-slope, y-slope]. Notice that the gradient has as many components as the input vector, rather than the number of coordinates of a point on the graph.
f := (x,y) -> x^2 + 2*y^2;
xval :=2: yval:= 3:
fx := diff(f(x,y),x);
xslope := eval(fx, {x=xval, y=yval});
fy := diff(f(x,y), y);
yslope := eval(fy, {x=xval, y=yval});
gradient := <xslope,yslope>;
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We are also told that the gradient should have direction that is perpendicular to the contour of the graph at that point. To see this put the contourplot and the gradient on the same graph.
cplot := contourplot(f(x,y),x=xval-2..xval+2,
y=yval-2..yval+2):
gradv := plot([xval+t*xslope,
yval+t*yslope,t=0..1],
thickness=3, color=green):
display({cplot, gradv},scaling=constrained);
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This relationship can be seen more clearly with the Maple command gradplot that plots a field of scaled gradient vectors.
gplot := gradplot(f(x,y),x=xval-2..xval+2,
y=yval-2..yval+2):
cplot := contourplot(f(x,y),x=xval-2..xval+2,
y=yval-2..yval+2):
display({cplot, gplot}, scaling=constrained);
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Tied into the idea that the gradient is perpendicular to the contours is that it is the direction of steepest ascent.
This means that we can visually check to see if we have computed the gradient correctly. If we take the cross section in the direction of the gradient it should be the direction that the function is rising most steeply.
f := (x,y) -> x^2 + 2*y^2;
xval :=2: yval:= 3:
zval := f(xval,yval):
fx := diff(f(x,y),x):
mx := eval(fx, {x=xval, y=yval}):
fy := diff(f(x,y), y):
my := eval(fy, {x=xval, y=yval}):
gradvec := <mx, my>;
unitgradient := gradvec/sqrt(gradvec.gradvec);
gradientderiv := mx*unitgradient[1]+my*unitgradient[2];
surf:= plot3d(f(x,y), x=xval-2..xval+2, y=yval-2..yval+2,
axes=boxed, color=red, style = patchcontour):
gradientpath :=spacecurve([xval+t*unitgradient[1], yval+t*unitgradient[2],
f(xval+t*unitgradient[1], yval+t*unitgradient[2])], t=-2..2, color=blue, thickness=2):
cutplane := plot3d([xval+u*unitgradient[1], yval+u*unitgradient[2],zval+v],
u=-2..2,v=-5..5, color=yellow, style=patchnogrid):
gradline :=spacecurve(
[xval+t*unitgradient[1], yval+t*unitgradient[2], zval+t*gradientderiv],
t=0..1, color=green, thickness=3):
display3d({cutplane, surf, gradientpath, gradline});
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A second look at directional derivatives
The alternate way to produce the directional derivative is to use the gradient, computing the partial derivatives, evaluate them at our point, and then take the sum of the partials times the appropriate component of the unit direction vector.
f := (x,y) -> x^2 + 2*y^2:
xval :=2: yval:= 3:
dirvec := <5, 12>:
uvec := dirvec/sqrt(dirvec.dirvec);
fx := diff(f(x,y),x);
xslope := eval(fx, {x=xval, y=yval});
fy := diff(f(x,y), y);
yslope := eval(fy, {x=xval, y=yval});
dirderiv := xslope*uvec[1]+yslope*uvec[2];
LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIoUyQpeic=
LCRJInhHNiIiIiM=
IiIl
LCRJInlHNiIiIiU=
IiM3
IyIkayIiIzg=
Notice that we do get the same answer for the directional derivative that we got with the formal definition above.
This second method is more roundabout, but it has the advantage of being easier to do when we don't have Maple handy. In the second method we only have to take limits for two cross sections.
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Comparing the two methods of finding directional derivatives
It is worthwhile verifying that the two methods of computing directional derivative give the same answer for any direction.
(To be more precise, that these two methods with produce the same answer in every direction is the definition of the function being differentiable at a point.)
We recall computations made above to get the x-slope and y-slope at a point.
f := (x,y) -> x^2 + 2*y^2:
xval :=2: yval:= 3:
dirvec := <3, 4>:
zval := f(xval,yval);
xslope := eval(diff(f(x,y), x), {x=xval, y=yval});
yslope := eval(diff(f(x,y), y), {x=xval, y=yval});
IiNB
IiIl
IiM3
We recall that a unit vector in direction theta has coordinates (cos(theta), sin(theta)). We then find a directional derivative in direction theta by using the definition of directional derivative.
ddera := (f,theta) -> limit((f(xval+t*cos(theta), yval+t*sin(theta))-f(xval, yval))/t, t=0);
Zio2JEkiZkc2IkkmdGhldGFHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JJmxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQqJiwmLTkkNiQsJkkleHZhbEdGJSIiIiomSSJ0R0YlRjctSSRjb3NHRiw2IzklRjdGNywmSSV5dmFsR0YlRjcqJkY5RjctSSRzaW5HRiVGPEY3RjdGNy1GMzYkRjZGPyEiIkY3RjlGRS9GOSIiIUYlRiVGJQ==
We now compute the directional derivative by the method that uses x-slope and y-slope.
dderb := theta -> cos(theta)*xslope + sin(theta)*yslope;
Zio2I0kmdGhldGFHNiJGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmKiYtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IzkkIiIiSSd4c2xvcGVHRiVGMkYyKiYtSSRzaW5HRiVGMEYySSd5c2xvcGVHRiVGMkYyRiVGJUYl
It is now a simple matter to graphically compare the values obtained by the two methods for all angles theta.
plot({ddera(f,theta)- dderb(theta)}, theta=0..2*Pi, axes=boxed);
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If we plot the ring of points (a+cos(theta), b+sin(theta), f(a,b)+ddera(theta)) we get a ring in the tangent plane (rotate the plot to see the underside of the tangent plane).
surf:= plot3d(f(x,y), x=xval-2..xval+2,
y=yval-2..yval+2,
axes=boxed, color=red, style = patchnogrid):
tanplane:= plot3d(zval+xslope*(x-xval)+
yslope*(y-yval),
x=xval-2..xval+2, y=yval-2..yval+2,
axes=boxed, color=yellow, style =
patchnogrid):
tanring :=spacecurve([xval+cos(theta),
yval+sin(theta),
f(xval,yval)+ddera(f,theta)],
theta=0..2*Pi, color=blue, thickness=2):
display3d({surf, tanplane, tanring});
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An example where the methods disagree
The problem with looking at nice examples where everything works is it makes one wonder why anyone came up with the definition of directional derivatives in the first place when the gradient method is so much easier. It is thus worth looking at a case when the partials all work, but the function is not differentiable.
The function we want to work with is 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 This looks like quite a strange function, but it looks a lot nicer in polar coordinates. Then we have 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 The point of this function is that in any direction through the origin it is well behaved, in fact it is linear. However the tangent lines through the origin do not turn into a plane.
rcos5theta := (x,y)-> piecewise(x = 0,0,(x^5-10*x^3*y^2+5*x*y^4)/(x^2+y^2)^2);
rcos5theta(x,y);
rcos5thetapolar := (r,theta) -> r*cos(5*theta);
rcos5thetapolar(r,theta);
Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYlLzkkIiIhRjAqJiwoKiRGLyIiJiIiIiomRi8iIiQ5JSIiIyEjNSomRi9GNUY4IiIlRjRGNSwmKiRGL0Y5RjUqJEY4RjlGNSEiI0YlRiVGJQ==
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plot3d(rcos5theta(x,y),x=-1..1,y=-1..1, axes=boxed, style=patchcontour);
plot3d([r,theta,rcos5thetapolar(r,theta)],r=0..1,theta=0..2*Pi,coords=cylindrical,axes=boxed,grid=[50,50],style=patchcontour);
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