Math 302 Learning Objectives Winter 2014 Multivariable Calculus (Part I) 13.1 Vectors in Three-Dimensional Space Plot points in three-dimensional space. Find the distance between two points in three-dimensional space. Write the equation of a sphere centered at a given point with a given radius. Find symmetric points about: a point, a line, or a plane. Parameterize a line segment. Find the midpoint of a line segment. 13.2 Vectors in Three-Dimensional Space Define vector. Represent a vector in its coordinate form, as a directed line segments and as a linear combination of unit vectors. Perform basic operations on vectors including: addition, subtraction, and scalar multiplication. Interpret these operations geometrically. Find the norm (magnitude or length) of a vector. Know and apply the norm properties of vectors. Determine whether two vectors are parallel. Find the unit vector in the direction of a given vector. Explain the natural correspondence between points and vectors. 13.3 The Dot Product Define dot product. Evaluate the dot product of two vectors. Know and apply the properties of dot product. Interpret the dot product geometrically. Use the dot product to determine if two vectors are perpendicular (or orthogonal). Use the dot product to find the angle between two vectors. Find the projection and component of projection of one vector onto another. Explain what these are geometrically. State and prove the Schwartz inequality. 13.4 The Cross Product Define cross product. Evaluate the cross product of two vectors. Know and apply the properties of cross product. Interpret the cross product geometrically. Use the cross product to find a vector perpendicular to two given nonparallel vectors. Use the cross product to find the area of a triangle or a parallelogram. Define scalar triple product. Use the scalar triple product to find the volume of a parallelepiped. State and prove Lagrange’s identity. 13.5 Lines Given information in a variety of ways, find the critical information needed to write the equation of a line; namely, a point and a direction vector. Find the equation of the line represented in any one of the following ways: o Vector parameterization o Scalar parametric equations o Symmetric form Determine whether a pair of lines is intersecting, parallel, or skew. Find the angle between two lines. Find the distance from a point to a line. 13.6 Planes Given information in a variety of ways, find the critical information needed to write the equation of a plane; namely, a point and a normal vector. Find the equation of the plane represented in either of the following ways: o Scalar equation o Vector equation o Symmetric form Determine whether a collection of vectors is coplanar. Find the distance from a point to a plane. 13.7 Higher dimensions Generalize the following for Rn. o The distance between two points. o The midpoint of a line segment. o The sum, difference and scalar multiples of vectors. o Dot product. o The norm of a vector. o The angle between two vectors. Linear Algebra 2.1 Systems of Linear Equations Define o linear equation o system of linear equations o solution o solution set Identify linear equations and systems of linear equations Relate the following types of solution sets of a system of two or three variables to the intersections of lines in a plane or the intersection of planes in three space: o a unique solution. o infinitely many solutions. o no solution. Represent a linear system as an augmented matrix and vice versa. Transform a system to a triangular pattern and then apply back substitution to solve the linear system. Represent the solution set to a linear system using parametric equations. 2.2 Direct Methods for Solving Linear Systems Identify matrices that are in row echelon form and reduced row echelon form. Determine whether a system of linear equations has no solution, a unique solution or an infinite number of solutions from its echelon form. Apply elementary row operations to transform systems of linear equations. Solve systems of linear equations using: o Gaussian elimination o Gauss-Jordan elimination Define and evaluate the rank of a matrix. Apply the Rank Theorem relate the rank of an augmented matrix to the solution set of a system in the case of: o homogeneous systems o nonhomogeneous systems Model and solve application problems using linear systems. 2.3 Spanning Sets and Linear Independence Explain what is meant by the span of a set of vectors both geometrically and algebraically. Determine the span of a set of vectors. Determine if a given vector is in the span of a set of vectors. Define linear independence. Determine whether a set of vectors is linearly dependent or linearly independent. For sets that are linearly dependent, determine a dependence relation. Prove theorems about span and linear independence. 3.1 Matrix Operations Perform the operations of: o matrix addition o scalar multiplication o transposition o matrix multiplication Represent matrices in terms of double subscript notation. Recall and demonstrate that the cancellation laws for scalar multiplication do not hold for matrix multiplication. Use matrices and matrix operations to model application problems. Represent and compute matrix products in terms of blocks. 3.2 Matrix Algebra Recall and apply the algebraic properties for o matrix addition o scalar multiplication on matrices o matrix multiplication o transposition Prove algebraic properties for matrices. Apply the concepts of span and linear independence to matrices. Recall that matrix multiplication is not commutative in general. Determine conditions under which matrices do commute. 3.3 The Inverse of a Matrix Define the inverse of a matrix. Recall the Fundamental Theorem of Invertible Matrices and the properties of invertible matrices. Prove theorems involving matrix inverses. Recall and apply the formula for the inverse of 2 × 2 matrices. Demonstrate the relationship between elementary matrices and row operations. Determine inverses of elementary matrices. Compute the inverse of a matrix using the Gauss-Jordan method. Solve a linear equation using matrix inverses. 3.4 The LU Factorization Find the LU factorization of a matrix. Use the LU factorization of a matrix to solve a system of linear equations. 3.5 Subspaces, Basis, Dimension, and Rank Define subspace Rn. Determine whether or not a given set of vectors forms a subspace of Rn. Define for a given matrix: o row space o column space o null space Determine whether or not a given vector is in a specified row space, column space or null space. Define for a subspace: o basis o dimension Given a subspace, determine its dimension and a basis. Verify whether or not a given set of vectors is a basis for the subspace. Define for a matrix: o rank o nullity Determine the rank and nullity of a given matrix. Prove and recall theorems involving the rank, nullity, and invertibility of matrices. 3.6 Linear Transformations Define linear transformation. Determine whether or not a given transformation is linear. Determine the matrix that represents a given linear transformation of vectors. Prove and recall theorems involving linear transformations. Find compositions and inverses of linear transformations. 3.7 Applications Model Markov processes. Determine the: o transition matrix o state vectors o probability vectors o steady state vectors Model population growth using a Leslie matrix. Investigate the behavior of the growth modeled by a Leslie matrix. 4.1 Introduction to Eigenvalues and Eigenvectors Interpret the eigenvalue problem: o algebraically o geometrically Determine whether a given vector is an eigenvector. Verify that a given value is an eigenvalue. Determine eigenvalues and eigenvectors: o geometrically o from the graph of the eigenspace Find the eigenvalues and eigenvectors of a general 2 × 2 matrix. 4.2 Determinants Apply the Laplace Expansion to evaluate determinants of n × n matrices. Recall and apply the properties of determinants to evaluate determinants, including: o det(AB) = det(A) det(B) o det(kA) = kn det(A) o det(A-1)= 1/det(A) o det(AT)=det(A) Recall the effects that row operations have on the determinants of matrices. Relate elementary matrices to row operations. Prove identities involving determinants. Evaluate matrix inverses using the adjoint method. Determine whether or not a matrix has an inverse based on its determinant. Use Cramer's rule to solve a linear system. 4.3 Eigenvalues and Eigenvectors of n × n Matrices Given an n × n matrix, compute o the characteristic polynomial o the Eigenvalues o a basis for each Eigenspace o the algebraic and geometric multiplicities of each Eigenvalue Solve application problems involving Eigenvalues and Eigenvectors. Recall and prove theorems involving Eigenvalues and Eigenvectors. 4.4 Similarity and Diagonalization Define similar matrices. Determine whether or not two matrices are similar. Find the diagonalization of a matrix or determine that the matrix is not diagonalizable. Find powers of a matrix using the diagonalization of a matrix. Prove theorems involving the similarity and diagonalization of matrices. Multivariable Calculus (Part II) 14.1 Limit, Continuity, Vector Derivative Define the following: o scalar functions o vector functions o components of a vector function o plane curve or space curve o parametrization of a curve o limit of a vector function o a vector function continuous at a point o derivative of a vector function o integral of a vector function Graph a parametric curve. Identify a curve given its parametrization. Determine combinations of vector functions such as sums, vector products and scalar products. Evaluate limits, derivatives, and integrals of vector functions. Recall, derive and apply rules for the following: o limits o differentiation o integration Determine continuity of a vector-valued function. 14.2 The Rules of Differentiation Recall, derive and apply differentiation rules to combinations of vector functions Prove theorems involving derivatives of vector-valued functions. Solve application problems involving derivatives of vector-valued functions. 14.3 Curves Define the following: o directed path o differentiable parameterized curve o tangent vector o tangent line o unit tangent vector o principal normal vector o normal line o osculating plane Find the tangent vector and tangent line to a curve at a given point. Find the principle normal and normal line to a curve at a given point. Determine the osculating plane for a space curve at a given point. Reverse the direction of a curve. Solve application problems involving curves. 14.4 Arc Length Define the following: o arc length o arc length parametrization Evaluate the arc length of a curve. Determine whether a curve is arc length parameterized. Find the arc length parametrization of a curve, if possible. 14.5 Curvilinear Motion; Curvature Define the following: o velocity vector function o speed o acceleration vector function o uniform circular motion o curvature o tangential component of acceleration o normal component of acceleration Given the position vector function of a moving object, calculate the velocity vector function, speed, and acceleration vector function, and vice versa. Calculate the curvature of a space curve. Recall the formulas for the curvature of a parameterized planar curve or a planar curve that is the graph of a function. Apply these formulas to calculate the curvature of a planar curve. Determine the tangential and normal components of acceleration for a given parameterized curve. Solve application problems involving curvilinear motion and curvature. 14.6 Vector Calculus in Mechanics Define the following: o force vector o momentum vector o angular momentum vector o torque Solve application problems involving force, momentum, angular momentum, and torque. 15.1 Elementary Examples Define the following: o real-valued function of several variables o domain o range o bounded functions Describe the domain and range of a function of several variables. Write a function of several variables given a description. 15.2 A Brief Catalogue of the Quadric Surfaces; Projections Define the following: o quadric surface o intercepts o traces o sections o symmetry o boundedness o cylinder o ellipsiod o elliptic cone o elliptic paraboloid o hyperboloid of one sheet o hyperboloid of two sheets o hyperbolic paraboloid o parabolic cylinder o elliptic cylinder o projection of a curve onto a coordinate plane Identify standard quadratic surfaces given their functions or graphs. Sketch the graph of a quadratic surface by sketching intercepts, traces, sections, centers, symmetry, boundedness. Find the projection of a curve, that is the intersection of two surfaces, to a coordinate plane. 15.3 Graphs; Level Curves and Level Surfaces Define the following: o level curve o level surface Describe the level sets of a function of several variables. Graphically represent a function of two variables by level curves or a function of three variables by level surfaces. Identify the characteristics of a function from its graph or from a graph of its level curves (or level surfaces). Solve application problems involving level sets. 15.4 Partial Derivatives Define the following: o partial derivative of a function of several variables o second partial derivative o mixed partial derivative Interpret the definition of a partial derivative of a function of two variables graphically. Evaluate the partial derivatives of a function of several variables. Evaluate the higher order partial derivatives of a function of several variables. Verify equations involving partial derivatives. Apply partial derivatives to solve application problems. 15.5 Open and Closed Sets Define the following: o neighborhood of a point o deleted neighborhood of a point o interior of a set o boundary of a set o open set o closed set Determine the boundary and interior of a set. Determine whether a set is open, closed, neither, or both. 15.6 Limits and Continuity; Equality of Mixed Partials Define the following: o limit of a function of several variables at a point o continuity of a function of several variables at a point Evaluate the limit of a function of several variables or show that it does not exists. Determine whether or not a function is continuous at a given point. Recall and apply the conditions under which mixed partial derivatives are equal. 16.1 Differentiability and Gradient Define the following: o differentiable multivariable function o gradient of a multivariable function Evaluate the gradient of a function. Find a function with a given gradient. 16.2 Gradients and Directional Derivatives Define the following: o directional derivative o isothermals Recall and prove identities involving gradients. Give a graphical interpretation of the gradient. Evaluate the directional derivative of a function. Give a graphical interpretation of directional derivative. Recall, prove, and apply the theorem that states that a differential function f increases most rapidly in the direction of the gradient (the rate of change equal to the length of the gradient) and it decreases most rapidly in the opposite direction (the rate of change equal to the opposite of the length of the gradient). Find the path of a heat seeking or a heat repelling particle. Solve application problems involving gradient and directional derivatives. 16.3 The Mean-Value Theorem; The Chain Rule Define the following: o the Mean Value Theorem for functions of several variables o normal line o chain rules for functions of several variables o implicit differentiation Recall and apply the Mean Value Theorem for functions of several variables and its corollaries. Apply an appropriate chain rule to evaluate a rate of change. Apply implicit differentiation to evaluate rates of change. Solve application problems involving chain rules and implicit differentiation. 16.4 The Gradient as a Normal; Tangent Lines and Tangent Planes Define the following: o normal vector o tangent vector o tangent line o tangent plane o normal line Use gradients to find the following for a smooth planar curve at a given point. o normal vector o normal line o tangent line or tangent plane Use gradients to find the following for a smooth surface at a given point. o normal vector o normal line o tangent plane Solve application problems involving normals and tangents to curves and surfaces. 16.5 Local Extreme Values Define the following: o local minimum and local maximum o critical points o stationary points o saddle points o discriminant o Second Derivative Test Find the critical points of a function of two variables. Apply the Second-Partials Test to determine whether each critical point is a local minimum, a local maximum, or a saddle point. Solve word problems involving local extreme values. 16.6 Absolute Extreme Values Define the following: o absolute minimum and absolute maximum o bounded subset of a plane or three-space o the Extreme Value Theorem Determine absolute extreme values of a function defined on a closed and bounded set. Apply the Extreme Value Theorem to justify the method for finding extreme values of functions defined on certain sets. Solve word problems involving absolute extreme values. 16.7 Maxima and Minima with Side Conditions Define the following: o side conditions or constraints o method of Lagrange o Lagrange multipliers o cross-product equation of the Lagrange condition Graphically interpret the method of Lagrange. Determine the extreme values of a function subject to a side conditions by applying the method of Lagrange. Apply the cross-product equation of the Lagrange condition to solve extreme value problems subject to side conditions. Apply the method of Lagrange to solve word problems. 16.8-9 Differentials; Reconstructing a Function from its Gradient Define the following: o differential o general solution o particular solution o connected open set o open region o simple closed curve o simply connected open region o partial derivative gradient test Determine the differential for a given function of several variables. Determine whether or not a vector function is a gradient. Given a vector function that is a gradient, find the functions with that gradient. 17.1-3 Multiple-Sigma Notation; The Double Integral over a Rectangle; The Evaluation of Double Integrals by Repeated Integrals Define the following: o double sigma notation o triple sigma notation o upper sum o lower sum o double integral o integral formula for the volume of a solid bounded between a region Ω in o the xy-plane and the graph of a non-negative function z=f(x,y) defined on Ω. o integral formula for the area of region in a plane o integral formula for the average of a function defined on a region Ω. o projection of a region onto a coordinate axis o Type I and Type II regions o reduction formulas for double integrals o the geometric interpretation of the reduction formulas for double integrals Evaluate double and triple sums given their sigma notation. Recall and apply summation identities. Approximate the integral of a function by a lower sum and an upper sum. Evaluate the integral of a function using the definition. Evaluate double integrals over a rectangle using the reduction formulas. Sketch planar regions and determine if they are Type I, Type II, or both. Evaluate double integrals over Type I and Type II regions. Change the order of integration of an integral. Apply double integrals to calculate volumes, areas, and averages. 17.4 The Double Integral as the Limit of Riemann Sums; Polar Coordinates Define the following: o diameter of a set o Riemann sum o double integral as a limit of Riemann sums o polar coordinates [r,ϴ] o transformation formulas between Cartesian and polar coordinates o double integral conversion formula between Cartesian and polar coordinates Represent a region in both Cartesian and polar coordinates. Evaluate double integrals in terms of polar coordinates. Evaluate areas and volumes using polar coordinates. Convert a double integral in Cartesian coordinates to a double integral in polar coordinates and then evaluate. 17.5 Further Applications of the Double Integral Define the following: o integral formula for the mass of a plate o integral formulas for the center of mass of a plate o integral formulas for the centroid of a plate o integral formulas for the moment of an inertia of a plate o radius of gyration o the Parallel Axis Theorem Evaluate the mass and center or mass of a plate Evaluate the centroid of a plate. Evaluate the moments of inertia of a plate. Calculate the radius of gyration of a plate. Recall and apply the parallel axis theorem. 17.6-7 Triple Integrals; Reduction to Repeated Integrals Define the following: o triple integral o integral formula for the volume of a solid o integral formula for the mass of a solid o integral formulas for the center of mass of a solid Evaluate physical quantities using triple integrals such as volume, mass, center of mass, and moments of intertia. Recall and apply the properties of triple integrals, including: linearity, order, additivity, and the mean-value condition. Sketch the domain of integration of an iterated integral. Change the order of integration of a triple integral. 17.8 Cylindrical Coordinates Define the following: o cylindrical coordinates of a point o coordinate transformations between Cartesian and cylindrical coordinates o cylindrical element of volume Convert between Cartesian and cylindrical coordinates. Describe regions in cylindrical coordinates. Evaluate triple integrals using cylindrical coordinates. 17.9 Spherical Coordinates Define the following: o spherical coordinates of a point o coordinate transformations between Cartesian and spherical coordinates o spherical element of volume Convert points and equations between Cartesian and spherical coordinates. Describe regions in spherical coordinates. Convert integrals in Cartesian coordinates to integrals in spherical coordinates. Set up and evaluate triple integrals using spherical coordinates. 17.10 Jacobians; Changing Variables in Multiple Integration Define the following: o Jacobian o change of variable formula for double integration o change of variable formula for triple integration Find the Jacobian of a coordinate transformation. Use a coordinate transformation to evaluate double and triple integrals. 18.1,4 Line Integrals; Another Notation for Line Integrals; Integrals with Respect to Arc Length Define the following: o work along a curved path o piecewise smooth curve Evaluate the work done by a varying force over a curved path. Evaluate line integrals in general including line integrals with respect to arc length. Evaluate the physical characteristics of a wire such as centroid, mass, and center of mass using line integrals. Determine whether or not a vector field is a gradient. Determine whether or not a differential form is exact. 18.2 The Fundamental Theorem for Line Integrals; Work-Energy Formula; Conservation of Mechanical Energy Define the following: o path-independent line integrals o gradient field Determine whether or not a force field is gradient field on a given region, and if so, find its potential function. Recall, apply, and verify the Fundamental Theorem for Line Integrals. Identify when the Fundamental Theorem of Line Integrals does not apply. 18.5 Green's Theorem Define the following: o Jordan curve o Jordan region o Green's Theorem Recall and verify Green's Theorem. Apply Green's Theorem to evaluate line integrals. Apply Green's Theorem to find the area of a region. Derive identities involving Green's Theorem 18.6 Parameterized Surfaces; Surface Area Define the following: o parameterized surface o fundamental vector product o element of surface area for a parameterized surface o surface integral o integral formula for the surface area of a parameterized surface o integral formula for the surface area of a surface = (, ) o upward unit normal Parameterize a surface. Evaluate the fundamental vector product for a parameterized surface. Calculate the surface area of a parameterized surface. Calculate the surface area of a surface = (, ). 18.7 Surface Integrals Define the following: o surface integral o integral formulas for the surface area and centroid of a parameterized surface o integral formulas for the mass and center of mass of a parameterized surface o integral formulas for the moments of inertia of a parameterized surface o integral formula for flux through a surface Calculate the surface area and centroid of a parameterized surface. Calculate the mass and center of mass of a parameterized surface. Calculate the moments of inertia of a parameterized surface. Evaluate the flux of a vector field through a surface. Solve application problems involving surface integrals. 18.8 The Vector Differential Operator ∇ Define the following: o the vector differential operator ∇ o divergence o curl o Laplacian Evaluate the divergence of a vector field. Evaluate the curl of a vector field Evaluate the Laplacian of a function. Recall, derive and apply formulas involving divergence, gradient and Laplacian. Interpret that divergence and curl of a vector fields physically. 18.9 The Divergence Theorem Define the following: o outward unit normal o the divergence theorem o sink and source o solenoidal Recall and verify the Divergence Theorem. Apply the Divergence Theorem to evaluate the flux through a surface. Solve application problems using the Divergence Theorem. 18.10 Stokes' Theorem Define the following: o oriented surface o outward, upward, and downward unit normal o the positive sense around the boundary of a surface o circulation o component of curl in the normal direction o irrotational o Stokes' theorem Recall and verify Stoke's theorem. Use Stokes' Theorem to calculate the flux of a curl vector field through a surface by a line integral. Apply Stoke's theorem to calculate the work (or circulation) of a vector field around a simple closed curve.