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Calculus III Study Guide
3-D Distance Formula
Equation of a Sphere:
With Center:
Vectors
Dot Product:
Cross Product:
Equations of Lines
Equation of Planes
Distance between point and plane
Quadric Surfaces
Ellipsoid:
Elliptic Paraboloid:
Hyperbolic Paraboloid:
Cone:
Hyperboloid of One Sheet:
Hyperboloid of two Sheets:
Vector Functions
If
Arc Length
Curvature
A parametrization is called smooth on an interval I if r’ is
continuous and on I. A curve is called smooth if it has a
smooth parametrization.
Unit Tangent Vector:
Normal Vector
Binormal Vector
Functions of Several Variables
Level Curves (Surfaces): Lowering the dimension by 1 by setting one
variable as a constant in order to view the “contour curves.”
Multivariable Limits
*If the limit exists, it must approach the same value from EVERY path in
the domain.*
Continuity
A function f of two variables is called continuous at (a,b) if
Partial Derivatives
Notations for Partial Derivatives:
Partial Differentiation Rule:
1. To find , regard y (or any other variables) as a constant and
differentiate with respect to x.
2. To find , regard x (or any other variables) as a constant and
differentiate with respect to y.
Tangent Planes and Linear Approximations
Equation of the tangent plane of f , continuous, to the surface
at the point :
Differentials
The Chain Rule
If
Implicit Differentiation
Directional Derivatives
The Directional Derivative of f at in the direction of a unit
vector :
Gradient
Maximizing the Directional Derivative
Theorem: If f is a function of two or three variables, then the maximum
value of the directional derivative is and it occurs
when has the same direction as the gradient vector .
Multivariable Max/Min
Theorem: If f has a local maximum or minimum at and the first-
order partial derivatives of f exist there, then
A point is called a critical point of f if , ,
or if one of these partial derivatives does not exist.
*At a multivariable critical point a function may have local max/min or
neither.*
Second Derivatives Test: Suppose the second partial derivatives of f
are continuous, and suppose that is a critical point of f:
a) If , then is a local minimum.
b) If , then is a local maximum.
c) If , then is not a local maximum or minimum.
*Case c) is called a saddle point of f and the graph of f crosses its
tangent plane at .
*If , we get no information; it could be any of the three.
Absolute Maximum and Minimum Values
Definition: A boundary point of D is a point such that every
disk with center contains points in D and also
points not in D.
Definition: A closed set in is one that contains all its boundary
points.
Definition: A bounded set in is one that is contained within
some disk. In other words, it is finite in extent.
Extreme Value Theorem:
If f is continuous on a closed, bounded set D in ,
then f attains an absolute maximum value
and an absolute minimum value at some
points and in D.
Method for Finding Absolute Maximums and Minimums
Must be a continuous function f on a closed, bounded set D:
1) Find the values of f at the critical points of f in D.
2) Find the extreme values of f on the boundary of D.
3) The largest of the values from steps 1 and 2 is the absolute
maximum value; the smallest of these values is the absolute
minimum value.
Lagrange Multipliers
If such that,
The number is called the Lagrange Multiplier.
Method of Lagrange Multipliers
To find the maximum and minimum values of subject to the
constraint [assuming that these extreme values exist and
on the surface :
(a)Find all values of such that
and
(b)Evaluate f at all points that result from step (a). The
largest of these values is the maximum value of f; the smallest is
the minimum value of f.
Two Constraints
Multiple Integrals
Double Integral of f over the rectangle R:
If
Midpoint Rule
Average Value
Partial Integration and Iterated Integrals
Fubini’s Theorem:
General Regions of Integration
Type 1:
Type 2:
Polar Coordinates
Where
If f is continuous on the polar region:
Triple Integrals
Fubini’s Theorem
Triple Integrals over General Bounded Region E
Type 1:
Subtype 1: Type I Plane Region
Subtype 2: Type II Plane Region
Type 2:
Subtype 1: Type I Plane Region
Subtype 2: Type II Plane Region
Type 3:
Subtype 1: Type I Plane Region
Subtype 2: Type II Plane Region
Triple Integrals in Cylindrical Coordinates
Triple Integrals in Spherical Coordinates
Change of Variables in Multiple Integrals
Consider a Transformation T from the uv-plane to the xy-plane:
Assume T is a Transformation, meaning g and have continuous first-
order partial derivatives.
Jacobian
Change of Variables in Double Integrals
Suppose T is a Transformation with a nonzero Jacobian, mapping a
region S in the uv-plane onto a region R in the xy-plane. Assuming f is
continuous on R, that R and S are of type I or II regions, and also that T
is one-to-one inside the boundary of S, then:
*
is the absolute value of the Jacobian*
* denotes the Jacobian of n variables.
Change of Variables in Triple Integrals
Assuming the same arguments as above:
Vector Calculus
Vector Fields
Definition: Let D be a set in (a n-dimensional region). A vector field
on is a function F that assigns to each point in D an n-
dimensional vector .
(In other words, a vector field is a space (or dimensions) of points, as
commonly known, outputting vectors at each point).
Gradient Fields
The gradient function is really just a vector field in n-
dimensional space.
Conservative Vector Field
A vector field F is called a conservative vector field if it is the gradient
of some scalar function, that is, if there exists a function f such
that . In this case f is called a potential function for F.
Line Integrals
Line Integrals of Vector Fields
*Fundamental Theorem of Line Integrals*
*This says that for a conservative vector field all we need to know is the
value of r at the endpoints to solve the line integral.*
Independence of Path
A curve is called closed if .
Theorem:
is independent of path in D iff
for
every closed path C in D.
An open set is a set that does not contain any of its boundary points.
(This means for every point there is a disk with center P that lies
entirely in the set.)
A set is connected if any two points in the set can be joined by a path
that lies in the set.
Theorem: Suppose is a vector field that is continuous on an open
connected region D. if
is independent of path in D,
then is a conservative vector field on D; that is, there
exists a function f such that .
A simple curve is a curve that doesn’t intersect itself anywhere
between its endpoints.
A simply-connected region is a connected region such that every simply
closed curve in the region encloses only points that are in the region.
Condition for Conservative Vector Field (Open Simply-Connected
Region)
*Green’s Theorem*
A positive orientation of a simple closed curve C refers to a single
counterclockwise traversal of C.
Let C be a positively oriented, piecewise-smooth, simple closed curve in
the plane and let D be the region bounded by C. If P and Q have
continuous partial derivatives on an open region that contains D, then:
Curl
Theorem: If f is a function of three variables that has continuous
second-order partial derivatives, then:
Theorem: If is a vector field defined on all of whose component
functions have continuous partial derivatives and
, then is a conservative vector field.
Divergence
Laplace Operator
Green’s First Identity
Green’s Second Identity
Parametric Surfaces
Surfaces of Revolution
Tangent Planes
Surface Area
Surface Integrals
If
If is a continuous vector field defined on an oriented surface S with
unit normal vector n, then the surface integral of F over S is:
(This is called flux)
*Stokes’ Theorem*
Let S be an oriented piecewise-smooth surface that is bounded by a
simple, closed, piecewise-smooth boundary curve C with positive
orientation.
*The Divergence Theorem*
Let E be a simple solid region and let S be the boundary surface of E,
given with positive (outward) orientation. Let be a vector field whose
component functions have continuous partial derivatives on an open
region that contains E.
Courtesy of
Allaire Mathematics
