Geometry of Rates of Changes of Surfaces and their Gradients, Levels Curves Calc IV Lab Karen Donnelly Saint Joseph's College

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRis= Load the plots package. QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIi Load the plottools package. QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJKnBsb3R0b29sc0dGJSEiIg== QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJL1ZlY3RvckNhbGN1bHVzR0YlISIi Load the Student MultivariateCalculus package QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMmSShTdHVkZW50R0YnNiNJNU11bHRpdmFyaWF0ZUNhbGN1bHVzR0YoISIi

Definition: The gradient vector for a function of two variables, f(x,y), is the vector 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. Properties: Some of the important properties of the gradient vector include: 1. It gives the direction of maximum increase of the function f at (x, y). 2. Its negative gives the direction of "minimum increase" of f at (x, y) -- i.e. the direction of "greatest descent") 3. It is normal to the level curve of f passing through (x, y).

Definition: Let f be a function of two variables x and y and let u = cos\316\270 i + sin \316\270 j be a unit vector The directional derivative of f at (x,y) in the direction of the vector u is given by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYnUSJ1RicvJSVib2xkR1EldHJ1ZUYnRjIvRjZRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRkEvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZERjU= f( x, y )= 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 provided this limit exists. Theorem: (Working definition for Directional Derivative). If f is a differentiable function of two variables, then the directional derivative in the direciton of the unit vector u at the point (x,y) is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYnUSJ1RicvJSVib2xkR1EldHJ1ZUYnRjIvRjZRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRkEvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZERjU= f( x, y ) = 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

1. Define the function PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmKiQ5JCIiI0YvKiQ5JUYvIiIkRiRGJEYk Plot the surface z = f(x,y) along with level curves: QyQ+SSNzcEc2Ii1JJ3Bsb3QzZEdGJTYpLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YsOy1JIi1HNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYzSShfc3lzbGliR0YlNiMiIiNGOi9GLUYvL0kmc3R5bGVHRiVJLVBBVENIQ09OVE9VUkdGJS9JKWNvbnRvdXJzR0YlIiM/L0klYXhlc0dGJUkmYm94ZWRHRiUvSSdsYWJlbHNHRiU3JUYsRi1JInpHRiUhIiI=SSNzcEc2Ig== 2. Compute the Gradient vector for 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 at [x,y] using the Gradient command from the Student[MultivariateCalculus] package: LUkpR3JhZGllbnRHNiI2JC1JImZHRiQ2JEkieEdGJEkieUdGJDckRilGKg== 3. Plot the gradient vector field for this function with the gradplot command from the plots library: QyQ+SSNncEc2Ii1JKWdyYWRwbG90R0YlNiktSSJmR0YlNiRJInhHRiVJInlHRiUvRiw7LUkiLUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRjNJKF9zeXNsaWJHRiU2IyIiI0Y6L0YtRi8vSShzY2FsaW5nR0YlSSxjb25zdHJhaW5lZEdGJS9JJ2Fycm93c0dGJUkmVEhJQ0tHRiUvSSVncmlkR0YlNyQiIzpGRS9JJmNvbG9yR0YlSSVibHVlR0YlISIiSSNncEc2Ig== 4. Plot some level curves for this function with the countourplot command: QyQ+SSNjcEc2Ii1JLGNvbnRvdXJwbG90R0YlNictSSJmR0YlNiRJInhHRiVJInlHRiUvRiw7LUkiLUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRjNJKF9zeXNsaWJHRiU2IyIiI0Y6L0YtRi8vSShzY2FsaW5nR0YlSSxjb25zdHJhaW5lZEdGJS9JKWNvbnRvdXJzR0YlIiM1ISIiSSNjcEc2Ig== 5. Display the gradient vectors and the level curves together: LUkoZGlzcGxheUc2IjYkPCRJI2NwR0YkSSNncEdGJC9JKHNjYWxpbmdHRiRJLGNvbnN0cmFpbmVkR0Yk 6. By using the transform command from the plottools library to transform both the countour plot and the gradient field to from the plane (2-space) to 3-space, we can then display the surface, countour plot, and level curves together: QyQ+SSV0cjNkRzYiLUkqdHJhbnNmb3JtR0YlNiNmKjYkSSJ4R0YlSSJ5R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiU3JTkkOSUhIiNGJUYlRiUhIiI=LUkoZGlzcGxheUc2IjYlPCVJI3NwR0YkLUkldHIzZEdGJDYjSSNjcEdGJC1GKTYjSSNncEdGJC9JJXZpZXdHRiQ3JTstSSItRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHNiRGNkkoX3N5c2xpYkdGJDYjIiIjRj1GMjtGMyIjPy9JKHNjYWxpbmdHRiRJLnVuY29uc3RyYWluZWRHRiQ= Question: What geometric relationships exist between the surface 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, its gradient vectors, and its level curves? LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwoKiY5JCIiIjklIiIjRi8tSSRjb3NHRiQ2I0YuRjFGLiEiIkYkRiRGJA== QyQ+SSNncEc2Ii1JKWdyYWRwbG90R0YlNigtSSJmR0YlNiRJInhHRiVJInlHRiUvRiw7LUkiLUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRjNJKF9zeXNsaWJHRiU2IyIiI0Y6L0YtRi8vSSdhcnJvd3NHRiVJJlRISUNLR0YlL0klZ3JpZEdGJTckIiM6RkIvSSZjb2xvckdGJUklYmx1ZUdGJSEiIg==SSNncEc2Ig== Just for fun we plot level curves with "filled=true" option and a coloring scheme of red to pink. Can you tell how the coloring works? QyQ+SSRjcGZHNiItSSxjb250b3VycGxvdEdGJTYoLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YsOy1JIi1HNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYzSShfc3lzbGliR0YlNiMiIiNGOi9GLUYvL0kpY29udG91cnNHRiUiIz8vSSdmaWxsZWRHRiVJJXRydWVHRjMvSSljb2xvcmluZ0dGJTckSSRyZWRHRiVJJXBpbmtHRiUhIiI=SSRjcGZHNiI= However, for display with gradient vector, it is better to not use the fill: QyQ+SSNjcEc2Ii1JLGNvbnRvdXJwbG90R0YlNictSSJmR0YlNiRJInhHRiVJInlHRiUvRiw7LUkiLUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRjNJKF9zeXNsaWJHRiU2IyIiI0Y6L0YtRi8vSSljb250b3Vyc0dGJSIjPy9JKWNvbG9yaW5nR0YlNyRJJWJsdWVHRiVJJmdyZWVuR0YlISIiSSNjcEc2Ig== LUkoZGlzcGxheUc2IjYkPCRJI2NwR0YkSSNncEdGJC9JKHNjYWxpbmdHRiRJLGNvbnN0cmFpbmVkR0Yk QyQ+SSNzcEc2Ii1JJ3Bsb3QzZEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYpLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YvOy1JIi1HNiRGKS9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzR0YoNiMiIiNGOi9GMEYyL0kmc3R5bGVHRiVJLVBBVENIQ09OVE9VUkdGJS9JKWNvbnRvdXJzR0YlIiM1L0klYXhlc0dGJUkmYm94ZWRHRiUvSSdsYWJlbHNHRiU3JUYvRjBJInpHRiUhIiI=SSNzcEc2Ig== QyQ+SSV0cjNkRzYiLUkqdHJhbnNmb3JtR0YlNiNmKjYkSSJ4R0YlSSJ5R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiU3JTkkOSUhIilGJUYlRiUhIiI=LUkoZGlzcGxheUc2IjYlPCVJI3NwR0YkLUkldHIzZEdGJDYjSSNjcEdGJC1GKTYjSSNncEdGJC9JJXZpZXdHRiQ3JTstSSItRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHNiRGNkkoX3N5c2xpYkdGJDYjIiIjRj1GMjstRjQ2IyIiKSIiJS9JKHNjYWxpbmdHRiRJLnVuY29uc3RyYWluZWRHRiQ=

The function in problem number 8 from section 13.6 PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJC1JJGV4cEdGJDYjLCYqJDkkIiIjISIiKiQ5JUYyRjNGJEYkRiQ= QyQ+SSNzcEc2Ii1JJ3Bsb3QzZEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYnLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YvOy1JIi1HNiRGKS9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzR0YoNiMiIiNGOi9GMEYyL0kmc3R5bGVHRiVJLVBBVENIQ09OVE9VUkdGJS9JJWF4ZXNHRiVJJmJveGVkR0YlISIiSSNzcEc2Ig== QyQ+SSNjcEc2Ii1JLGNvbnRvdXJwbG90R0YlNictSSJmR0YlNiRJInhHRiVJInlHRiUvRiw7LUkiLUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRjNJKF9zeXNsaWJHRiU2IyIiI0Y6L0YtRi8vSShzY2FsaW5nR0YlSSxjb25zdHJhaW5lZEdGJS9JKWNvbnRvdXJzR0YlIiM1ISIiSSNjcEc2Ig== QyQ+SSNncEc2Ii1JKWdyYWRwbG90R0YlNiktSSJmR0YlNiRJInhHRiVJInlHRiUvRiw7LUkiLUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRjNJKF9zeXNsaWJHRiU2IyIiI0Y6L0YtRi8vSShzY2FsaW5nR0YlSSxjb25zdHJhaW5lZEdGJS9JJ2Fycm93c0dGJUkmVEhJQ0tHRiUvSSVncmlkR0YlNyQiIzpGRS9JJmNvbG9yR0YlSSVibHVlR0YlISIiSSNncEc2Ig== LUkoZGlzcGxheUc2IjYjPCRJI2NwR0YkSSNncEdGJA== QyQ+SSV0cjNkRzYiLUkqdHJhbnNmb3JtR0YlNiNmKjYkSSJ4R0YlSSJ5R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiU3JTkkOSUhIiJGJUYlRiVGMw==LUkoZGlzcGxheUc2IjYlPCVJI3NwR0YkLUkldHIzZEdGJDYjSSNjcEdGJC1GKTYjSSNncEdGJC9JJXZpZXdHRiQ3JTstSSItRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHNiRGNkkoX3N5c2xpYkdGJDYjIiIjRj1GMjstRjQ2IyIiIkZBL0koc2NhbGluZ0dGJEkudW5jb25zdHJhaW5lZEdGJA==

When we have a function of three variables, we would need 4-space to represent this function w = f(x,y,z). To assist in visualization of the behavior of the function we can use its gradient vectors and its level surfaces. Consider the following function: PkkiZkc2ImYqNiVJInhHRiRJInlHRiRJInpHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwoKiQ5JCIiIyIiIiokOSVGMEYxOSYhIiVGJEYkRiQ= Compute its gradient vector. LUkpR3JhZGllbnRHNiI2JC1JImZHRiQ2JUkieEdGJEkieUdGJEkiekdGJDclRilGKkYr We can use the gradplot3d to plot the gradient field for this function: QyQ+SSNnZEc2Ii1JK2dyYWRwbG90M2RHRiU2KS1JImZHRiU2JUkieEdGJUkieUdGJUkiekdGJS9GLDstSSItRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiRGNEkoX3N5c2xpYkdGJTYjIiIjRjsvRi1GMC9GLjtGMSIiJS9JJWdyaWRHRiU3JSIiJkZDRkMvSSZjb2xvckdGJUkmYmxhY2tHRiUvSSVheGVzR0YlSSZib3hlZEdGJSIiIg==SSNnZEc2Ig== We use the implicitplot3d command to plot a couple of level surfaces of this function. QyQ+SSRsczFHNiItSS9pbXBsaWNpdHBsb3QzZEdGJTYpLy1JIitHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYtSShfc3lzbGliR0YlNiQtRis2JCokSSJ4R0YlIiIjLUkiKkdGLDYkRjgqJEkieUdGJUY4LUkiLUdGLDYjLUY6NiQiIiVJInpHRiUtRj82IyIiIi9GNzstRj82I0Y4RjgvRj1GSS9GRDtGSkZDL0kmc3R5bGVHRiVJKndpcmVmcmFtZUdGJS9JJWF4ZXNHRiVJJmJveGVkR0YlL0koc2NhbGluZ0dGJUksY29uc3RyYWluZWRHRiUhIiI=SSRsczFHNiI= QyQ+SSRsczJHNiItSS9pbXBsaWNpdHBsb3QzZEdGJTYpLy1JIitHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YlSS9WZWN0b3JDYWxjdWx1c0c2JEYtSShfc3lzbGliR0YlNiQtRis2JCokSSJ4R0YlIiIjLUkiKkdGLDYkRjgqJEkieUdGJUY4LUkiLUdGLDYjLUY6NiQiIiVJInpHRiUiIikvRjc7LUY/NiNGOEY4L0Y9RkcvRkQ7RkhGQy9JJnN0eWxlR0YlSSp3aXJlZnJhbWVHRiUvSSVheGVzR0YlSSZib3hlZEdGJS9JKHNjYWxpbmdHRiVJLGNvbnN0cmFpbmVkR0YlISIiSSRsczJHNiI= Displaying the gradient field with these level surfaces gives an indication of their geometrical relationship (see below). Questions: What geometric relationships exist between a function of three variables, its level surfaces, and its gradient vectors? LUkoZGlzcGxheUc2IjYlPCVJI2dkR0YkSSRsczFHRiRJJGxzMkdGJC9JJWF4ZXNHRiRJJmJveGVkR0YkL0koc2NhbGluZ0dGJEksY29uc3RyYWluZWRHRiQ=

This package (used for the Gradient command above) has a number of features useful for multivariable calculus. For example we can use the GradientTutor command to investigate some of the concepts of gradients -- this is from the Example worksheet found in Help -- Example Worksheets - Calculus LUkuR3JhZGllbnRUdXRvckc2IjYkLUkiK0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiRJL1ZlY3RvckNhbGN1bHVzRzYkRilJKF9zeXNsaWJHRiQ2JC1JIipHRig2JEkieUdGJCokSSJ4R0YkIiIjKiRGM0Y2LzckRjVGMzckNyQiIiJGNjckIiIkIiIm After executing the above statement you see in the bottom it gives the Maple Gradient command that would be executed to generate the plot -- following this syntax as an example we can generate a similar plot for the function 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. LUkpR3JhZGllbnRHNiI2JS1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYpSShfc3lzbGliR0YkNiQtRic2JCIiIy1JJHNpbkdGLTYjSSJ4R0YkLUkkY29zR0YtNiNJInlHRiQvNyRGNkY6NyQ3JCIiIkY/NyQtSSItR0YoNiNGP0Y/L0knb3V0cHV0R0YkSSVwbG90R0Yt QyU+SSNndkc2Ii1JKUdyYWRpZW50R0YlNiQtSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJUkvVmVjdG9yQ2FsY3VsdXNHNiRGLEkoX3N5c2xpYkdGJTYkLUYqNiQiIiMtSSRzaW5HRjA2I0kieEdGJS1JJGNvc0dGMDYjSSJ5R0YlLzckRjlGPTckIiIiRkFGQS1JJmV2YWxmR0YsNiNJIiVHRiU= We can add some plot options if desired: LUkpR3JhZGllbnRHNiI2Ki1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEYpSShfc3lzbGliR0YkNiQtRic2JCIiIy1JJHNpbkdGLTYjSSJ4R0YkLUkkY29zR0YtNiNJInlHRiQvNyRGNkY6NyQ3JCIiIkY/NyQtSSItR0YoNiNGP0Y/L0Y2Oy1GQjYjIiImRkgvRjpGRS9JInpHRiQ7LUZCNiMkIiNEISIiRk8vSSdvdXRwdXRHRiRJJXBsb3RHRi0vSShzY2FsaW5nR0YkSSxjb25zdHJhaW5lZEdGJC9JJWF4ZXNHRiRJJmJveGVkR0Yk

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= From plots: contourplot implicitplot3d gradplot gradplot3d From plottools:: transform From VectorCalculus Gradient From Student[MultivariateCalculus] Gradient GradientTutor