Chapter 1

Parametric Equations and Polar Coordinates

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Summary

Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve

Key terms

parametric equations

the equations x = x ( t ) and y = y ( t ) that define a parametric curve

angular coordinate

θ the angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial ( x ) axis, measured counterclockwise

parameter

an independent variable that both x and y depend on in a parametric curve; usually represented by the variable t

parametric curve

the graph of the parametric equations x ( t ) and y ( t ) over an interval a ≤ t ≤ b combined with the equations

cardioid

a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid is r = a ( 1 + sin θ ) or r = a…

directrix

a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two

discriminant

the value 4 A C - B 2 , which is used to identify a conic when the equation contains a term involving x y , is called a discriminant

eccentricity

the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix

Chapter 2

Vectors in Space

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Summary

We can add vectors by using the parallelogram method or the triangle method to find the sum. The magnitude of a vector is a scalar: ‖ v ‖ = x 2 + y 2 A unit vector u has magnitude 1 and can be found by dividing a vector by its magnitude: u = 1 ‖ v . The standard unit vectors are i = 〈 1 , 0 〉 and j = 〈 0 , 1 〉 .

Key terms

vector

a mathematical object that has both magnitude and direction

parallelogram method

a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of…

triangle method

a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a…

component

a scalar that describes either the vertical or horizontal direction of a vector

standard unit vectors

unit vectors along the coordinate axes: i = 〈 1 , 0 〉 , j = 〈 0 , 1 〉

magnitude of a vector

a scalar: ‖ v ‖ = x 2 + y 2

unit vector

a vector with margnitude 1

Chapter 3

Vector-Valued Functions

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Summary

To calculate the limit of a vector-valued function, calculate the limits of the component functions separately To calculate the derivative of a vector-valued function, calculate the derivatives of the component functions, then put them back into a new vector-valued function The derivative of a vector-valued function r ( t ) is also a tangent vector to the curve. A vector-valued function is a function of the form r ( t ) = f ( t ) i + g ( t ) j or r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k , r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k , where the component functions f, g, and…

Key terms

vector-valued function

a function of the form r ( t ) = f ( t ) i + g ( t ) j or r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k , r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k , where the component…

component functions

the component functions of the vector-valued function r ( t ) = f ( t ) i + g ( t ) j are f ( t ) and g ( t ) , and the component functions of the vector-valued function r ( t )…

derivative of a vector-valued function

the derivative of a vector-valued function r ( t ) is r ′ ( t ) = lim Δ t → 0 r ( t + Δ t ) - r ( t ) Δ t , r ′ ( t ) = lim Δ t → 0 r ( t + Δ t ) - r ( t ) Δ t , provided the…

plane curve

the set of ordered pairs ( f ( t ) , g ( t ) together with their defining parametric equations x = f ( t ) and y = g ( t )

space curve

the set of ordered triples ( f ( t ) , g ( t ) , h ( t ) together with their defining parametric equations x = f ( t ) , y = g ( t ) and z = h ( t )

limit of a vector-valued function

a vector-valued function r ( t ) has a limit L as t approaches a if lim t → a | r ( t ) - L | = 0

tangential component of acceleration

the coefficient of the unit tangent vector T when the acceleration vector is written as a linear combination of T and N

Chapter 4

Differentiation of Functions of Several Variables

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Summary

The graph of a function of two variables is a surface in ℝ 3 and can be studied using level curves and vertical traces A set of level curves is called a contour map To study limits and continuity for functions of two variables, we use a δ disk centered around a given point A function of several variables has a limit if for any point in a δ ball centered at a point P , the value of the function at that point is arbitrarily close to a fixed value (the limit value)

Key terms

partial derivative

a derivative of a function of more than one independent variable in which all the variables but one are held constant

vertical trace

the set of ordered triples ( c , y , z ) that solves the equation f ( c , y ) = z for a given constant x = c or the set of ordered triples ( x , d , z ) that solves the equation…

function of two variables

a function z = f ( x , y ) that maps each ordered pair ( x , y ) in a subset D of ℝ 2 to a unique real number z

δ ball

all points in ℝ 3 lying at a distance of less than δ from ( x 0 , y 0 , z 0 )

contour map

a plot of the various level curves of a given function f ( x , y )

surface

the graph of a function of two variables, z = f ( x , y )

δ disk

an open disk of radius δ centered at point ( a , b )

Chapter 5

Multiple Integration

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Summary

We can use Fubini’s theorem to write and evaluate a double integral as an iterated integral We can use a double Riemann sum to approximate the volume of a solid bounded above by a function of two variables over a rectangular region. By taking the limit, this becomes a double integral representing the volume of the solid Properties of double integral are useful to simplify computation and find bounds on their values

Key terms

Fubini’s theorem

if f ( x , y ) is a function of two variables that is continuous over a rectangular region R = { ( x , y ) ∈ ℝ 2 | a ≤ x ≤ b , c ≤ y ≤ d } , R = { ( x , y ) ∈ ℝ 2 | a ≤ x ≤ b , c…

Jacobian

the Jacobian J ( u , v ) in two variables is a 2 × 2 determinant: J ( u , v ) = | ∂ x ∂ u ∂ y ∂ u ∂ x ∂ v ∂ y ∂ v | ; J ( u , v ) = | ∂ x ∂ u ∂ y ∂ u ∂ x ∂ v ∂ y ∂ v | ; the…

one-to-one transformation

a transformation T : G → R defined as T ( u , v ) = ( x , y ) is said to be one-to-one if no two points map to the same image point

polar rectangle

the region enclosed between the circles r = a and r = b and the angles θ = α and θ = β ; it is described as R = { ( r , θ ) | a ≤ r ≤ b , α ≤ θ ≤ β } R = { ( r , θ ) | a ≤ r ≤ b…

triple integral

the triple integral of a continuous function f ( x , y , z ) over a rectangular solid box B is the limit of a Riemann sum for a function of three variables, if this limit exists

Chapter 6

Vector Calculus

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Summary

A vector field F is called conservative if there exists a scalar function f such that ∇ f = F Line integrals generalize the notion of a single-variable integral to higher dimensions. The domain of integration in a single-variable integral is a line segment along the x -axis, but the domain of integration in a line integral is a curve in a plane or in space A vector field assigns a vector F ( x , y ) to each point ( x , y ) in a subset D of ℝ 2 or ℝ 3 .

Key terms

single-variable integral

a line segment along the x -axis, but the domain of integration in a line integral is a curve in a plane or in space

vector field

measured in ℝ 2 , an assignment of a vector F ( x , y ) to each point ( x , y ) of a subset D of ℝ 2 ; in ℝ 3 , an assignment of a vector F ( x , y , z ) to each point ( x , y …

vector field F

called conservative if there exists a scalar function f such that ∇ f = F

line integral

the integral of a function along a curve in a plane or in space

circulation

the tendency of a fluid to move in the direction of curve C . If C is a closed curve, then the circulation of F along C is line integral ∫ C F · T d s , which we also denote ∫ C…

closed curve

a curve for which there exists a parameterization r ( t ) , a ≤ t ≤ b , such that r ( a ) = r ( b ) , and the curve is traversed exactly once

curl

the curl of vector field F = 〈 P , Q , R 〉 , denoted ∇ × F , is the “determinant” of the matrix | i j k ∂ x ∂ y ∂ z P Q R | and is given by the expression ( R y - Q z ) i + ( P z…

divergence

the divergence of a vector field F = 〈 P , Q , R 〉 , denoted ∇ ⋅ F , is P x + Q y + R z ; it measures the “outflowing-ness” of a vector field

Chapter 7

Second-Order Differential Equations

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Summary

To solve a nonhomogeneous linear second-order differential equation, first find the general solution to the complementary equation, then find a particular solution to the nonhomogeneous equation To find a general solution for a homogeneous second-order differential equation, we must find two linearly independent solutions. If y 1 ( x ) and y 2 ( x ) are linearly independent solutions to a second-order, linear, homogeneous differential equation, then the general solution is given by y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) . The form of the general solution varies depending on whether the characteristic equation has distinct, real roots; a single, repeated real root; or complex conjugate roots

Key terms

linearly independent

a set of functions f 1 ( x ) , f 2 ( x ) ,…, f n ( x ) f 1 ( x ) , f 2 ( x ) ,…, f n ( x ) for which there are no constants c 1 , c 2 ,… c n , such that c 1 f 1 ( x ) + c 2 f 2 (…

boundary conditions

the conditions that give the state of a system at different times, such as the position of a spring-mass system at two different times

characteristic equation

the equation a λ 2 + b λ + c = 0 for the differential equation a y ″ + b y ′ + c y = 0

particular solution

a solution y p ( x ) of a differential equation that contains no arbitrary constants

general solution

given by y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x )

linearly dependent

a set of functions f 1 ( x ) , f 2 ( x ) ,…, f n ( x ) f 1 ( x ) , f 2 ( x ) ,…, f n ( x ) for which there are constants c 1 , c 2 ,… c n , not all zero, such that c 1 f 1 ( x )…

method of variation of parameters

a method that involves looking for particular solutions in the form y p ( x ) = u ( x ) y 1 ( x ) + v ( x ) y 2 ( x ) , y p ( x ) = u ( x ) y 1 ( x ) + v ( x ) y 2 ( x ) , where…