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Chapter 5

Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus

Book Version 1
By Boundless
Boundless Calculus
Calculus
by Boundless
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Section 1
Vectors and the Geometry of Space
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Three-Dimensional Coordinate Systems

The three-dimensional coordinate system expresses a point in space with three parameters, often length, width and depth ($x$, $y$, and $z$).

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Vectors in the Plane

Vectors are needed in order to describe a plane and can give the direction of all dimensions in one vector equation.

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Vectors in Three Dimensions

A Euclidean vector is a geometric object that has magnitude (i.e. length) and direction.

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The Dot Product

The dot product takes two vectors of the same dimension and returns a single value.

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The Cross Product

The cross product of two vectors is a vector which is perpendicular to both of the original vectors.

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Equations of Lines and Planes

A line is a vector which connects two points on a plane and the direction and magnitude of a line determine the plane on which it lies.

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Cylinders and Quadric Surfaces

A quadric surface is any $D$-dimensional hypersurface in $(D+1)$-dimensional space defined as the locus of zeros of a quadratic polynomial.

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Cylindrical and Spherical Coordinates

Cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.

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Surfaces in Space

A surface is a two-dimensional, topological manifold.

Section 2
Vector Functions
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Vector-Valued Functions

A vector function covers a set of multidimensional vectors at the intersection of the domains of $f$, $g$, and $h$.

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Arc Length and Speed

Arc length and speed are, respectively, a function of position and its derivative with respect to time.

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Calculus of Vector-Valued Functions

A vector function is a function that can behave as a group of individual vectors and can perform differential and integral operations.

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Arc Length and Curvature

The curvature of an object is the degree to which it deviates from being flat and can be found using arc length.

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Planetary Motion According to Kepler and Newton

Kepler explained that the planets move in an ellipse around the Sun, which is at one of the two foci of the ellipse.

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Tangent Vectors and Normal Vectors

A vector is normal to another vector if the intersection of the two form a 90-degree angle at the tangent point.

Section 3
Partial Derivatives
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Functions of Several Variables

Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable.

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Limits and Continuity

A study of limits and continuity in multivariable calculus yields counter-intuitive results not demonstrated by single-variable functions.

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Partial Derivatives

A partial derivative of a function of several variables is its derivative with respect to a single variable, with the others held constant.

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Tangent Planes and Linear Approximations

The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.

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The Chain Rule

For a function $U$ with two variables $x$ and $y$, the chain rule is given as $\frac{d U}{dt} = \frac{\partial U}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial U}{\partial y} \cdot \frac{dy}{dt}$.

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Directional Derivatives and the Gradient Vector

The directional derivative represents the instantaneous rate of change of the function, moving through $\mathbf{x}$ with a velocity specified by $\mathbf{v}$.

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Maximum and Minimum Values

The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.

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Lagrange Multiplers

The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints.

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Optimization in Several Variables

To solve an optimization problem, formulate the function $f(x,y, \cdots )$ to be optimized and find all critical points first.

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Applications of Minima and Maxima in Functions of Two Variables

Finding extrema can be a challenge with regard to multivariable functions, requiring careful calculation.

Section 4
Multiple Integrals
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Double Integrals Over Rectangles

For a rectangular region $S$ defined by $x$ in $[a,b]$ and $y$ in $[c,d]$, the double integral of a function $f(x,y)$ in this region is given as $\int_c^d(\int_a^b f(x,y) dx) dy$.

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Iterated Integrals

An iterated integral is the result of applying integrals to a function of more than one variable.

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Double Integrals Over General Regions

Double integrals can be evaluated over the integral domain of any general shape.

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Double Integrals in Polar Coordinates

When domain has a cylindrical symmetry and the function has several specific characteristics, apply the transformation to polar coordinates.

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Triple Integrals in Cylindrical Coordinates

When the function to be integrated has a cylindrical symmetry, it is sensible to integrate using cylindrical coordinates.

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Triple Integrals in Spherical Coordinates

When the function to be integrated has a spherical symmetry, change the variables into spherical coordinates and then perform integration.

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Triple Integrals

For $T \subseteq R^3$, the triple integral over $T$ is written as $\iiint_T f(x,y,z)\, dx\, dy\, dz$.

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Change of Variables

One makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae.

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Applications of Multiple Integrals

Multiple integrals are used in many applications in physics and engineering.

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Center of Mass and Inertia

The center of mass for a rigid body can be expressed as a triple integral.

Section 5
Vector Calculus
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Vector Fields

A vector field is an assignment of a vector to each point in a subset of Euclidean space.

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Conservative Vector Fields

A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.

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Line Integrals

A line integral is an integral where the function to be integrated is evaluated along a curve.

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Fundamental Theorem for Line Integrals

Gradient theorem says that a line integral through a gradient field can be evaluated from the field values at the endpoints of the curve.

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Green's Theorem

Green's theorem gives relationship between a line integral around closed curve $C$ and a double integral over plane region $D$ bounded by $C$.

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Curl and Divergence

The four most important differential operators are gradient, curl, divergence, and Laplacian.

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Parametric Surfaces and Surface Integrals

A parametric surface is a surface in the Euclidean space $R^3$ which is defined by a parametric equation.

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Surface Integrals of Vector Fields

The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field.

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Stokes' Theorem

Stokes' theorem relates the integral of the curl of a vector field over a surface to the line integral of the field around the boundary.

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The Divergence Theorem

The divergence theorem relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.

Section 6
Second-Order Linear Equations
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Second-Order Linear Equations

A second-order linear differential equation has the form $\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)$, where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.

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Nonhomogeneous Linear Equations

Nonhomogeneous second-order linear equation are of the the form: $\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)$, where $f(t)$ is nonzero.

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Applications of Second-Order Differential Equations

A second-order linear differential equation can be commonly found in physics, economics, and engineering.

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Series Solutions

The power series method is used to seek a power series solution to certain differential equations.

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Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
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