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Several Variables Calculus
Section 1: Functions of Several Variables
Section 2: Limits and Continuity
Section 3: Partial Derivatives
Section 4: Tangent Planes and Linear Approximations
Section 5: The Chain Rule
Section 6: Directional Derivatives and the Gradient Vector
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Section 4: Variation of Parameters
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Section 8: Homogeneous Linear Systems with Constant Coefficients
Section 9: Complex Eigenvalues
Section 10: Fundamental Matrices
Section 11: Repeated Eigenvalues
Section 12: Nonhomogeneous Linear Systems
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Section 1: Probabilistic Models and Probability Laws
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Section 7: Joint Distribution Function, Marginal Probability Mass Function, and Uniform Distribution
Section 8: Independence of Two Random Variables, Covariance, and Correlation
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Several Variables Calculus
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Section 1: Functions of Several Variables
Section 1: Functions of Several Variables
Instructions
First, you should watch the concepts videos below explaining the topics in the section.
Second, you should attempt to solve the exercises and then watch the videos explaining the exercises.
Last, you should attempt to answer the self-assessment questions to determine how well you learned the material.
When you have finished the material below, you can start on
Section 2
or return to the
main several variable calculus page
.
Concepts
The definition of a function of two variables
The graph of a function of two variables with domain \(D\) and range \(R\)
The level curves of a function of two variables
Links & Resources
Download Notes
Return to Main Calculus Page
Return to Mini-Course Main Page
Watch Concepts Video 1
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Functions of Several Variables Conceptual V1
Watch Concepts Video 2
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Functions of Several Variables Conceptual V2
Exercises
Directions:
You should attempt to solve the problems first and then watch the video to see the solution.
Consider \(f(x,y)=\ln(y-4x)\).
Evaluate \(f(-1,3).\)
Find the domain of \(f(x,y)\) and sketch the domain of \(f(x,y)\) in the \(xy\)-plane.
Reveal Answer
\(f(-1,3)=\ln (7)\)
Domain: \( \{ (x,y) \mid y-4x>0\}\). See the video for a sketch.
Watch Video
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Functions of Several Variables Exercise V1
Find the domain of \(f(x,y)=\dfrac{\sqrt{9-x^2-y^2}}{x+y}\) and sketch the domain of \(f(x,y)\) in the \(xy\)-plane.
Reveal Answer
Domain: \( \left\{ (x,y) \mid 9-x^2-y^2\geq 0 \; \text{ and } \; x+y\neq 0\right\}\). See the video for a sketch.
Watch Video
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Functions of Several Variables Exercise V2
Sketch the level curves for \(f(x,y)=1-5x+y\) for \(k=1,0,-1.\)
Reveal Answer
See the video for the sketch.
Watch Video
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Functions of Several Variables Exercise V3
Sketch the level curves for \(f(x,y)=\sqrt{4-x^2-y^2}\) for \(k=0,1,2.\)
Reveal Answer
See the video for the sketch.
Watch Video
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Functions of Several Variables Exercise V4
Sketch the level curve for \(f(x,y)=\sqrt{y^2-x^2}\) for \(k=4.\)
Reveal Answer
See the video for the sketch.
Watch Video
If you would like to see more videos on this topic, click the following link and see the related videos. Note the related videos at the link are not required viewing.
Functions of Several Variables Exercise V5
Self-Assessment Questions
Directions:
The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
What are the necessary steps when finding the domain of a surface \(z=f(x,y)\)?
When finding the domain of a surface \(z=f(x,y)\), how do we determine the set of all points \((x,y)\) where \(f(x,y)\) is
not
defined?
How do the level curves of a surface \(z=f(x,y)\) help us visualize, and hence piece together, the graph of a surface?
What is the relation between a level curve and a horizontal trace?
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