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Thomas Calculus Multivariable 15th Edition By George Brinton Thomas, Joel R. Hass

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Pages: 777
File size: 49 MB
Format: Digital PDF
Course: Education
Category: eBook[PDF]

About this ebook

The Multivariable volume of Thomas' Calculus, 15th Edition, co-authored by Joel R. Hass, Christopher E. Heil, and Przemyslaw Bogacki (based on the original work by George Brinton Thomas), covers advanced multi-dimensional calculus typically taught in a Calculus III college course.

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The volume features a distinct emphasis on visualization, vector applications, and optimization techniques (including a modern look at gradient descent used in machine learning).

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The primary chapters and detailed topics covered in this edition include:

Quizlet

Chapter 12: Vectors and the Geometry of Space

Three-Dimensional Coordinate Systems: Plotting and distance formulas in

R3R-3
ℝ3
.
Vectors: Vector algebra, components, magnitude, unit vectors, and scalar multiplication.
The Dot Product: Angle between vectors, vector projections, and work calculations.
The Cross Product: Torques, areas of parallelograms, and triple scalar products (volume of parallelepipeds).
Lines and Planes in Space: Parametric, vector, and symmetric equations of lines; scalar and linear equations of planes.
Cylinders and Quadric Surfaces: Identifying and sketching surfaces like spheres, ellipsoids, paraboloids, and cones.
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Chapter 13: Vector-Valued Functions and Motion in Space

Vector-Valued Functions: Defining space curves, limits, derivatives, and integrals of vector functions.
Modeling Projectile Motion: Position, velocity, acceleration, and tracking paths in space.
Arc Length and Unit Tangent Vector (

Tbold cap T

): Finding the distance along a curved trajectory.
Curvature and the Principal Normal Vector (

Nbold cap N

): Quantifying how sharply a curve turns.
Tangential and Normal Components of Acceleration: Breaking acceleration into directions along and perpendicular to motion.

Chapter 14: Partial Derivatives

Functions of Several Variables: Domain, range, graphs, level curves, and level surfaces.
Limits and Continuity in Higher Dimensions: Analyzing behavior near multi-variable points.
Partial Derivatives: First-order and higher-order differentiation with respect to independent variables.
The Chain Rule: Application to composite multi-variable functions.
Directional Derivatives and Gradient Vectors: Gradients, maximal rate of change, and geometric tangent planes.
Extreme Values and Saddle Points: Local/absolute maxima and minima, the Second Derivative Test, and critical points.
Lagrange Multipliers: Optimization subject to equality constraints.
Taylor’s Formula for Two Variables: Polynomial approximations of multi-variable surfaces.
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Chapter 15: Multiple Integrals

Double and Iterated Integrals: Integrating over rectangular and non-rectangular regions in the

xyx y

-plane.
Area, Moments, and Center of Mass: Using double integrals to find physical properties of thin plates.
Double Integrals in Polar Form: Simplifying integration over circular or angular bounds.
Triple Integrals in Rectangular Coordinates: Calculating volumes and mass properties of 3D solids.
Triple Integrals in Cylindrical and Spherical Coordinates: Harnessing symmetry to evaluate complex 3D integration regions.
Substitutions in Multiple Integrals: Utilizing the Jacobian determinant for changing variables in multiple dimensions.
Quizlet

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Chapter 16: Integrals and Vector Fields

Line Integrals: Integrating scalar functions or vector fields along a path in space.
Vector Fields and Line Integrals: Computing work, circulation, and flux along a curve.
Path Independence, Conservative Fields, and Potential Functions: Testing for exactness and finding potential functions.
Green’s Theorem in the Plane: Relating line integrals around closed paths to double integrals over bounded regions.
Surfaces and Area: Parameterizing surfaces and computing surface areas.
Surface Integrals: Evaluating scalar and vector integrals over curved surfaces (flux across a surface).
The Divergence Theorem (Gauss's Theorem): Equating net flux out of a 3D solid to a triple integral of divergence.
Stokes’ Theorem: Equating the line integral around a 3D loop to a surface integral of curl.
Quizlet

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Appendix Additions (15th Edition Unique)

Determinants and Gradient Descent: Introduces core linear algebra tools and optimization math tailored directly for modern fields like Machine Learning and Neural Networks.
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