Skip to Content
OpenStax Logo
Calculus Volume 3

Key Concepts

Calculus Volume 3Key Concepts
Buy book
  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

2.1 Vectors in the Plane

  • Vectors are used to represent quantities that have both magnitude and direction.
  • We can add vectors by using the parallelogram method or the triangle method to find the sum. We can multiply a vector by a scalar to change its length or give it the opposite direction.
  • Subtraction of vectors is defined in terms of adding the negative of the vector.
  • A vector is written in component form as v=x,y.v=x,y.
  • The magnitude of a vector is a scalar: v=x2+y2.v=x2+y2.
  • A unit vector uu has magnitude 11 and can be found by dividing a vector by its magnitude: u=1vv.u=1vv. The standard unit vectors are i=1,0andj=0,1.i=1,0andj=0,1. A vector v=x,yv=x,y can be expressed in terms of the standard unit vectors as v=xi+yj.v=xi+yj.
  • Vectors are often used in physics and engineering to represent forces and velocities, among other quantities.

2.2 Vectors in Three Dimensions

  • The three-dimensional coordinate system is built around a set of three axes that intersect at right angles at a single point, the origin. Ordered triples (x,y,z)(x,y,z) are used to describe the location of a point in space.
  • The distance dd between points (x1,y1,z1)(x1,y1,z1) and (x2,y2,z2)(x2,y2,z2) is given by the formula
    d=(x2x1)2+(y2y1)2+(z2z1)2.d=(x2x1)2+(y2y1)2+(z2z1)2.
  • In three dimensions, the equations x=a,y=b,andz=cx=a,y=b,andz=c describe planes that are parallel to the coordinate planes.
  • The standard equation of a sphere with center (a,b,c)(a,b,c) and radius rr is
    (xa)2+(yb)2+(zc)2=r2.(xa)2+(yb)2+(zc)2=r2.
  • In three dimensions, as in two, vectors are commonly expressed in component form, v=x,y,z,v=x,y,z, or in terms of the standard unit vectors, xi+yj+zk.xi+yj+zk.
  • Properties of vectors in space are a natural extension of the properties for vectors in a plane. Let v=x1,y1,z1v=x1,y1,z1 and w=x2,y2,z2w=x2,y2,z2 be vectors, and let kk be a scalar.
    • Scalar multiplication: kv=kx1,ky1,kz1kv=kx1,ky1,kz1
    • Vector addition: v+w=x1,y1,z1+x2,y2,z2=x1+x2,y1+y2,z1+z2v+w=x1,y1,z1+x2,y2,z2=x1+x2,y1+y2,z1+z2
    • Vector subtraction: vw=x1,y1,z1x2,y2,z2=x1x2,y1y2,z1z2vw=x1,y1,z1x2,y2,z2=x1x2,y1y2,z1z2
    • Vector magnitude: v=x12+y12+z12v=x12+y12+z12
    • Unit vector in the direction of v: vv=1vx1,y1,z1=x1v,y1v,z1v,vv=1vx1,y1,z1=x1v,y1v,z1v, v0v0

2.3 The Dot Product

  • The dot product, or scalar product, of two vectors u=u1,u2,u3u=u1,u2,u3 and v=v1,v2,v3v=v1,v2,v3 is u·v=u1v1+u2v2+u3v3.u·v=u1v1+u2v2+u3v3.
  • The dot product satisfies the following properties:
    • u·v=v·uu·v=v·u
    • u·(v+w)=u·v+u·wu·(v+w)=u·v+u·w
    • c(u·v)=(cu)·v=u·(cv)c(u·v)=(cu)·v=u·(cv)
    • v·v=v2v·v=v2
  • The dot product of two vectors can be expressed, alternatively, as u·v=uvcosθ.u·v=uvcosθ. This form of the dot product is useful for finding the measure of the angle formed by two vectors.
  • Vectors u and v are orthogonal if u·v=0.u·v=0.
  • The angles formed by a nonzero vector and the coordinate axes are called the direction angles for the vector. The cosines of these angles are known as the direction cosines.
  • The vector projection of v onto u is the vector projuv=u·vu2u.projuv=u·vu2u. The magnitude of this vector is known as the scalar projection of v onto u, given by compuv=u·vu.compuv=u·vu.
  • Work is done when a force is applied to an object, causing displacement. When the force is represented by the vector F and the displacement is represented by the vector s, then the work done W is given by the formula W=F·s=Fscosθ.W=F·s=Fscosθ.

2.4 The Cross Product

  • The cross product u×vu×v of two vectors u=u1,u2,u3u=u1,u2,u3 and v=v1,v2,v3v=v1,v2,v3 is a vector orthogonal to both uu and v.v. Its length is given by u×v=u·v·sinθ,u×v=u·v·sinθ, where θθ is the angle between uu and v.v. Its direction is given by the right-hand rule.
  • The algebraic formula for calculating the cross product of two vectors,
    u=u1,u2,u3andv=v1,v2,v3,u=u1,u2,u3andv=v1,v2,v3, is
    u×v=(u2v3u3v2)i(u1v3u3v1)j+(u1v2u2v1)k.u×v=(u2v3u3v2)i(u1v3u3v1)j+(u1v2u2v1)k.
  • The cross product satisfies the following properties for vectors u,v,andw,u,v,andw, and scalar c:c:
    • u×v=(v×u)u×v=(v×u)
    • u×(v+w)=u×v+u×wu×(v+w)=u×v+u×w
    • c(u×v)=(cu)×v=u×(cv)c(u×v)=(cu)×v=u×(cv)
    • u×0=0×u=0u×0=0×u=0
    • v×v=0v×v=0
    • u·(v×w)=(u×v)·wu·(v×w)=(u×v)·w
  • The cross product of vectors u=u1,u2,u3u=u1,u2,u3 and v=v1,v2,v3v=v1,v2,v3 is the determinant |ijku1u2u3v1v2v3|.|ijku1u2u3v1v2v3|.
  • If vectors uu and vv form adjacent sides of a parallelogram, then the area of the parallelogram is given by u×v.u×v.
  • The triple scalar product of vectors u,u, v,v, and ww is u·(v×w).u·(v×w).
  • The volume of a parallelepiped with adjacent edges given by vectors u,v,andwu,v,andw is V=|u·(v×w)|.V=|u·(v×w)|.
  • If the triple scalar product of vectors u,v,andwu,v,andw is zero, then the vectors are coplanar. The converse is also true: If the vectors are coplanar, then their triple scalar product is zero.
  • The cross product can be used to identify a vector orthogonal to two given vectors or to a plane.
  • Torque ττ measures the tendency of a force to produce rotation about an axis of rotation. If force FF is acting at a distance rr from the axis, then torque is equal to the cross product of rr and F:F: τ=r×F.τ=r×F.

2.5 Equations of Lines and Planes in Space

  • In three dimensions, the direction of a line is described by a direction vector. The vector equation of a line with direction vector v=a,b,cv=a,b,c passing through point P=(x0,y0,z0)P=(x0,y0,z0) is r=r0+tv,r=r0+tv, where r0=x0,y0,z0r0=x0,y0,z0 is the position vector of point P.P. This equation can be rewritten to form the parametric equations of the line: x=x0+ta,x=x0+ta, y=y0+tb,y=y0+tb, and z=z0+tc.z=z0+tc. The line can also be described with the symmetric equations xx0a=yy0b=zz0c.xx0a=yy0b=zz0c.
  • Let LL be a line in space passing through point PP with direction vector v.v. If QQ is any point not on L,L, then the distance from QQ to LL is d=PQ×vv.d=PQ×vv.
  • In three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew.
  • Given a point PP and vector n,n, the set of all points QQ satisfying equation n·PQ=0n·PQ=0 forms a plane. Equation n·PQ=0n·PQ=0 is known as the vector equation of a plane.
  • The scalar equation of a plane containing point P=(x0,y0,z0)P=(x0,y0,z0) with normal vector n=a,b,cn=a,b,c is a(xx0)+b(yy0)+c(zz0)=0.a(xx0)+b(yy0)+c(zz0)=0. This equation can be expressed as ax+by+cz+d=0,ax+by+cz+d=0, where d=ax0by0cz0.d=ax0by0cz0. This form of the equation is sometimes called the general form of the equation of a plane.
  • Suppose a plane with normal vector n passes through point Q.Q. The distance DD from the plane to point PP not in the plane is given by
    D=projnQP=|compnQP|=|QP·n|n.D=projnQP=|compnQP|=|QP·n|n.
  • The normal vectors of parallel planes are parallel. When two planes intersect, they form a line.
  • The measure of the angle θθ between two intersecting planes can be found using the equation: cosθ=|n1·n2|n1n2,cosθ=|n1·n2|n1n2, where n1n1 and n2n2 are normal vectors to the planes.
  • The distance DD from point (x0,y0,z0)(x0,y0,z0) to plane ax+by+cz+d=0ax+by+cz+d=0 is given by
    D=|a(x0x1)+b(y0y1)+c(z0z1)|a2+b2+c2=|ax0+by0+cz0+d|a2+b2+c2.D=|a(x0x1)+b(y0y1)+c(z0z1)|a2+b2+c2=|ax0+by0+cz0+d|a2+b2+c2.

2.6 Quadric Surfaces

  • A set of lines parallel to a given line passing through a given curve is called a cylinder, or a cylindrical surface. The parallel lines are called rulings.
  • The intersection of a three-dimensional surface and a plane is called a trace. To find the trace in the xy-, yz-, or xz-planes, set z=0,x=0,ory=0,z=0,x=0,ory=0, respectively.
  • Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the form Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.
  • To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.
  • Important quadric surfaces are summarized in Figure 2.87 and Figure 2.88.

2.7 Cylindrical and Spherical Coordinates

  • In the cylindrical coordinate system, a point in space is represented by the ordered triple (r,θ,z),(r,θ,z), where (r,θ)(r,θ) represents the polar coordinates of the point’s projection in the xy-plane and zz represents the point’s projection onto the z-axis.
  • To convert a point from cylindrical coordinates to Cartesian coordinates, use equations x=rcosθ,x=rcosθ, y=rsinθ,y=rsinθ, and z=z.z=z.
  • To convert a point from Cartesian coordinates to cylindrical coordinates, use equations r2=x2+y2,r2=x2+y2, tanθ=yx,tanθ=yx, and z=z.z=z.
  • In the spherical coordinate system, a point PP in space is represented by the ordered triple (ρ,θ,φ),(ρ,θ,φ), where ρρ is the distance between PP and the origin (ρ0),(ρ0), θθ is the same angle used to describe the location in cylindrical coordinates, and φφ is the angle formed by the positive z-axis and line segment OP,OP, where OO is the origin and 0φπ.0φπ.
  • To convert a point from spherical coordinates to Cartesian coordinates, use equations x=ρsinφcosθ,x=ρsinφcosθ, y=ρsinφsinθ,y=ρsinφsinθ, and z=ρcosφ.z=ρcosφ.
  • To convert a point from Cartesian coordinates to spherical coordinates, use equations ρ2=x2+y2+z2,ρ2=x2+y2+z2, tanθ=yx,tanθ=yx, and φ=arccos(zx2+y2+z2).φ=arccos(zx2+y2+z2).
  • To convert a point from spherical coordinates to cylindrical coordinates, use equations r=ρsinφ,r=ρsinφ, θ=θ,θ=θ, and z=ρcosφ.z=ρcosφ.
  • To convert a point from cylindrical coordinates to spherical coordinates, use equations ρ=r2+z2,ρ=r2+z2, θ=θ,θ=θ, and φ=arccos(zr2+z2).φ=arccos(zr2+z2).
Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution-NonCommercial-ShareAlike License 4.0 and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction
Citation information

© Mar 30, 2016 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.