Chapter 5
Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
By Boundless
![Thumbnail](../../../../../figures.boundless-cdn.com/17794/square/3-20d-20cartisean-20system.jpg)
The three-dimensional coordinate system expresses a point in space with three parameters, often length, width and depth (
![Thumbnail](../../../../../figures.boundless-cdn.com/17797/square/planes-20in-20space.jpeg)
Vectors are needed in order to describe a plane and can give the direction of all dimensions in one vector equation.
![Thumbnail](../../../../../figures.boundless-cdn.com/18008/raw/3d-vector.jpg)
A Euclidean vector is a geometric object that has magnitude (i.e. length) and direction.
![Thumbnail](../../../../../figures.boundless-cdn.com/17799/square/dot-20product.jpg)
The dot product takes two vectors of the same dimension and returns a single value.
![Thumbnail](../../../../../figures.boundless-cdn.com/17800/square/right-20hand-20rule.jpg)
The cross product of two vectors is a vector which is perpendicular to both of the original vectors.
![Thumbnail](../../../../../figures.boundless-cdn.com/17973/square/vertical-line.jpg)
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.
![Thumbnail](../../../../../figures.boundless-cdn.com/18012/raw/circular-cylinder-rh.jpg)
A quadric surface is any
![Thumbnail](../../../../../figures.boundless-cdn.com/18010/raw/coord-system-cy-1.jpg)
Cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
![Thumbnail](../../../../../figures.boundless-cdn.com/18009/raw/sphere-wireframe.jpg)
A surface is a two-dimensional, topological manifold.
![Thumbnail](../../../../../figures.boundless-cdn.com/30581/square/vector-valued-function-2.jpg)
A vector function covers a set of multidimensional vectors at the intersection of the domains of
![Thumbnail](../../../../../figures.boundless-cdn.com/17743/square/arclength-approx.jpg)
Arc length and speed are, respectively, a function of position and its derivative with respect to time.
![Thumbnail](../../../../../figures.boundless-cdn.com/17740/square/vector-20.jpg)
A vector function is a function that can behave as a group of individual vectors and can perform differential and integral operations.
![Thumbnail](../../../../../figures.boundless-cdn.com/17744/square/radius-20of-20curvature.jpg)
The curvature of an object is the degree to which it deviates from being flat and can be found using arc length.
![Thumbnail](../../../../../figures.boundless-cdn.com/17720/square/kepler-second-law.gif)
Kepler explained that the planets move in an ellipse around the Sun, which is at one of the two foci of the ellipse.
![Thumbnail](../../../../../figures.boundless-cdn.com/17793/square/surface-20normal.jpg)
A vector is normal to another vector if the intersection of the two form a 90-degree angle at the tangent point.
![Thumbnail](../../../../../figures.boundless-cdn.com/17883/square/surface-plot.jpg)
Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable.
![Thumbnail](../../../../../figures.boundless-cdn.com/17882/square/ple-of-continuous-function.jpg)
A study of limits and continuity in multivariable calculus yields counter-intuitive results not demonstrated by single-variable functions.
![Thumbnail](../../../../../figures.boundless-cdn.com/17881/square/x2-2bx-2b1.jpg)
A partial derivative of a function of several variables is its derivative with respect to a single variable, with the others held constant.
![Thumbnail](../../../../../figures.boundless-cdn.com/17886/raw/image-tangent-plane.jpg)
The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.
![Thumbnail](../../../../../figures.boundless-cdn.com/17887/square/scalar-field.jpg)
For a function
![Thumbnail](../../../../../figures.boundless-cdn.com/17888/square/gradient99.jpg)
The directional derivative represents the instantaneous rate of change of the function, moving through
![Thumbnail](../../../../../figures.boundless-cdn.com/17889/square/saddle-point.jpg)
The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.
![Thumbnail](../../../../../figures.boundless-cdn.com/29391/square/nontangential.jpg)
The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints.
![Thumbnail](../../../../../figures.boundless-cdn.com/17893/raw/cuboid-simple.jpg)
To solve an optimization problem, formulate the function
![Thumbnail](../../../../../figures.boundless-cdn.com/17894/square/fig1.jpeg)
Finding extrema can be a challenge with regard to multivariable functions, requiring careful calculation.
![Thumbnail](../../../../../figures.boundless-cdn.com/17897/square/volume-under-surface.jpg)
For a rectangular region
![Thumbnail](../../../../../figures.boundless-cdn.com/17896/square/volume-under-surface.jpg)
An iterated integral is the result of applying integrals to a function of more than one variable.
![Thumbnail](../../../../../figures.boundless-cdn.com/17903/raw/mpio-formulediriduzione-r2.jpg)
Double integrals can be evaluated over the integral domain of any general shape.
![Thumbnail](../../../../../figures.boundless-cdn.com/18220/raw/minio-da-cartesiano-polare.jpg)
When domain has a cylindrical symmetry and the function has several specific characteristics, apply the transformation to polar coordinates.
![Thumbnail](../../../../../figures.boundless-cdn.com/17905/raw/cylindrical-coordinates.jpg)
When the function to be integrated has a cylindrical symmetry, it is sensible to integrate using cylindrical coordinates.
![Thumbnail](../../../../../figures.boundless-cdn.com/17906/raw/colatitude-2c-longitude-29.jpg)
When the function to be integrated has a spherical symmetry, change the variables into spherical coordinates and then perform integration.
![Thumbnail](../../../../../figures.boundless-cdn.com/17908/raw/-normalit-c3-a0-r3-esempio.jpg)
For
![Thumbnail](../../../../../figures.boundless-cdn.com/18271/raw/cylindrical-coordinates.jpg)
One makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae.
![Thumbnail](../../../../../figures.boundless-cdn.com/17909/raw/stribution-xy-line-segment.jpg)
Multiple integrals are used in many applications in physics and engineering.
![Thumbnail](../../../../../figures.boundless-cdn.com/17910/square/orbit3.gif)
The center of mass for a rigid body can be expressed as a triple integral.
![Thumbnail](../../../../../figures.boundless-cdn.com/17866/raw/vectorfield.jpg)
A vector field is an assignment of a vector to each point in a subset of Euclidean space.
![Thumbnail](../../../../../figures.boundless-cdn.com/17872/square/e-integral-of-scalar-field.gif)
A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
![Thumbnail](../../../../../figures.boundless-cdn.com/23733/square/e-integral-of-scalar-field.gif)
A line integral is an integral where the function to be integrated is evaluated along a curve.
![Thumbnail](../../../../../figures.boundless-cdn.com/17873/raw/ge-charge-plane-horizontal.jpg)
Gradient theorem says that a line integral through a gradient field can be evaluated from the field values at the endpoints of the curve.
![Thumbnail](../../../../../figures.boundless-cdn.com/18401/raw/-27s-theorem-simple-region.jpg)
Green's theorem gives relationship between a line integral around closed curve
![Thumbnail](../../../../../figures.boundless-cdn.com/17875/square/fig1.jpeg)
The four most important differential operators are gradient, curl, divergence, and Laplacian.
![Thumbnail](../../../../../figures.boundless-cdn.com/17878/raw/stokes-27-theorem.jpg)
A parametric surface is a surface in the Euclidean space
![Thumbnail](../../../../../figures.boundless-cdn.com/23735/raw/stokes-27-theorem.jpg)
The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field.
![Thumbnail](../../../../../figures.boundless-cdn.com/23736/raw/stokes-27-theorem.jpg)
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.
![Thumbnail](../../../../../figures.boundless-cdn.com/17879/raw/aceswithandwithoutboundary.jpg)
The divergence theorem relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.
![Thumbnail](../../../../../figures.boundless-cdn.com/17857/raw/simple-gravity-pendulum.jpg)
A second-order linear differential equation has the form
![Thumbnail](../../../../../figures.boundless-cdn.com/17860/square/heat-diffusion.jpg)
Nonhomogeneous second-order linear equation are of the the form:
![Thumbnail](../../../../../figures.boundless-cdn.com/17859/square/wo-pole-feedback-amplifier.jpg)
A second-order linear differential equation can be commonly found in physics, economics, and engineering.
![Thumbnail](../../../../../figures.boundless-cdn.com/17863/square/exp-series.gif)
The power series method is used to seek a power series solution to certain differential equations.