https://github.com/k-karna/multivariable_calculus
Solutions to Multivariable Calculus by Don Shimamoto | Solved step by step through LaTeX
https://github.com/k-karna/multivariable_calculus
multivariable-calculus real-valued-functions
Last synced: 11 months ago
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Solutions to Multivariable Calculus by Don Shimamoto | Solved step by step through LaTeX
- Host: GitHub
- URL: https://github.com/k-karna/multivariable_calculus
- Owner: k-karna
- Created: 2023-10-21T20:02:26.000Z (over 2 years ago)
- Default Branch: main
- Last Pushed: 2024-01-31T19:39:53.000Z (over 2 years ago)
- Last Synced: 2024-01-31T20:38:50.699Z (over 2 years ago)
- Topics: multivariable-calculus, real-valued-functions
- Language: Jupyter Notebook
- Homepage:
- Size: 628 KB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
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Metadata Files:
- Readme: README.md
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README
Multivariable Calculus
Author: Don Shimamoto
### I: Preliminaries
### II: Vector-valued function of one variable
#### Chapter 2: Path and Curves
- Parametrizations
- Velocity, acceleration, speed, arclength
- Intgrals with respect to arclength
- The geometry of curves: tangent and normal vectors
- The cross product
- The geometry of space curves: Frenet vectors
- Curvature and torsion
- The Frenet-Serret formulas
- The classification of space curves
- **Exercise for Chapter 2**
### III: Real-Valued Functions
#### Chapter 3: Real-valued functions: Preliminaries
- Graphs and level sets
- More surface in $\mathbb{R}^3$
- The equation of a plane in $\mathbb{R}^3$
- Open sets
- Continuity
- Some properties of continuous functions
- The Cauchy-Schwarz and triangle inequalities
- Limits
- **Exercise for chapter 3**
#### Chapter 4: Real-valued functions: Differentiation
- The first-order approximation
- Conditions for differentiability
- The mean value theorem
- The $C^1$ test
- The Little Chain Rule
- Directional derivatives
- $\nabla f$ as normal vector
- Higher-order partial derivatives
- Smooth functions
- Max/Min: Critical points
- Classifying nondegenrate critical points
- Max/Min: Lagrange Multipliers
- **Exercise for chapter 4**
#### Chapter 5: Real-valued fuctions: Integration
- Volume and iterated integrals
- The double integral
- Interpretations of the double integral
- Parametrization of surfaces
- Polar Coordinates $(r, \theta)$ in $\mathbb{R}^2$
- Cylindrical coordinates $(r, \theta, z)$ in $\mathbb{R}^3$
- Spherical coordinates $(p, \phi , \theta)$ in $\mathbb{R}^3$
- Integrals with respect to surface area
- Triple integrals and beyond
- **Exercise for chapter 5**
### IV: Vector-valued Functions
#### Chapter 6: Differentiability and the Chain rule
- Continuity revisited
- Differentiability revisited
- The chain rule : a conceptual approach
- The chain rule : a computational approach
- **Exercise for chapter 6**
#### Chapter 7: Change of Variables
- Change of vaiables for double integrals
- A word about substitution
- Examples : Linear changes of variables, symmetry
- Change of variables for $n-$fold integrals
- **Exercise for chapter 7**