Math 302 Outcome Statements Winter 2013

Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a plane (d) vector (e) components of a vector (f) vector addition (g) scalar multiplication (h) zero vector (i) vector subtraction (j) vector norm (magnitude, length) (k) unit vector (l) coordinate unit vectors i, j, k (m) linear combination of unit vectors B. Plot points in three-dimensional space. C. Calculate the distance between two points in two-dimensional space and 3-dimensional space D. Write the equation of a sphere centered about a given point with a given radius. Determine the center and radius of a sphere. given its equation. E. Write the component equations of a line that passes through two given points. F. Write the component equations of a line segment with given endpoints. G. Find the midpoint of a given line segment. H. Find the points of symmetry about a point, line, or plane. I. Represent a vector by each of the following: (a) components (b) a linear combination of coordinate unit vectors J. Carry out the vector operations: (a) addition (b) scalar multiplication (c) subtraction K. Represent the operations of vector addition, scalar multiplication and norm geometrically. L. Find the norm (magnitude, length) of a vector. Determine whether two vectors are parallel. M. Recall, apply and verify the basic properties of vector addition, scalar multiplication and norm. N. Model and solve application problems using vectors. Reading: Multivariable Calculus 13.1-2 2 The Dot Product (a) dot product. (b) perpendicular vectors. (c) unit vector in the direction of a vector a, denoted u a. (d) the projection of a on b, denoted proj b a.. (e) the b-component of a, denoted comp b a. (f) the direction cosines of a vector. (g) the direction angles of a vector. (h) the Schwarz Inequality. (i) the work done by a constant force on an object. (j) the dot product test for perpendicular vectors. (k) the dot product test for parallel vectors. (l) geometric interpretation of the dot product

B. Evaluate a dot product from the coordinate formula or the angle formula. C. Interpret the dot product geometrically. D. Evaluate the following using the dot product: (a) the length of a vector. (b) the angle between two vectors. (c) u a, the unit vector in the direction of a vector a. (d) proj b a, the projection of a on b. (e) comp b a, the b-component of a. (f) the direction cosines of a vector. (g) the direction angles of a vector. (h) the work done by a constant force on an object. E. Prove and verify the Schwarz Inequality. F. Prove and apply the dot product tests for perpendicular and parallel vectors. G. Recall and apply the properties of the dot product. H. Prove identities involving the dot product. I. Solve application problems involving the dot product. J. Extend the vector operations and related identities for addition, scalar multiplication, and dot product to higher dimensions. Reading: Multivariable Calculus 13.3,7 3 The Cross Product (a) the cross product of two vectors (b) scalar triple product B. Evaluate a cross product from the the coordinate formula or angle formula. C. Interpret the cross product geometrically. D. Evaluate the following using the cross product: (a) a vector perpendicular to two given vectors. (b) the area of a parallelogram. (c) the area or a triangle. (d) moment of force or moment of torque. E. Evaluate scalar triple products. F. Use the scalar triple product to determine the following: (a) volume of a parallelepiped. (b) whether or not three vectors are coplanar. G. Recall and apply the properties of the cross product and scalar triple product. H. Prove identities involving the cross product and the scalar triple product. I. Solve application problems involving the cross product and scalar triple product. Reading: Multivariable Calculus 13.4 4 Lines (a) direction vector for a line (b) vector equation of a line (c) scalar parametric equations of a line (d) Cartesian equations or symmetric form of a line B. Represent a line in 3-space by: (a) a vector equation (b) scalar parametric equations (c) Cartesian equations C. Find the equation(s) representing a line given information about

(a) a point of the line and the direction of the line or (b) two points contained in the line. (c) a point and a parallel line. (d) a point and perpendicular to a plane. (e) two planes intersecting in the line. D. Find the distance from a point to a line. E. Solve application problems involving lines. Reading: Multivariable Calculus 13.5 5 Planes (a) normal vector to a plane (b) cartesian equation of a plane (c) parametric equation of a plane B. Find the equation of a plane in 3-space given a point and a normal vector, three points, or a geometric description of the plane. C. Determine a normal vector and the intercepts of a given plane. D. Represent a plane by parametric equations. E. Find the distance from a point to a plane. F. Find the angle between a line and a plane. G. Determine a point of intersection between a line and a surface. H. Sketch planes given their equations. I. Solve application problems involving planes. Reading: Multivariable Calculus 13.6 6 Systems of Linear Equations (a) linear system of m equations in n unknowns (b) consistent and inconsistent (c) solution set (d) coefficient matrix (e) elementary row operations B. Identify linear systems. C. Represent a system of linear equations as an augmented matrix and vice versa. D. 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: i. a unique solution. ii. infinitely many solutions. iii. no solution. Reading: Linear Algebra 1.1 7 Gaussian elimination (a) reduced row echelon form (b) leading variables or pivots (c) free variables

(d) row echelon form (e) back substitution (f) Gaussian elimination (g) Gauss-Jordan elimination (h) homogeneous (i) trivial solution (j) nontrivial solutions B. Identify matrices that are in row echelon form and reduced row echelon form. C. Determine whether a linear system is consistent or inconsistent from its reduced row echelon form. If the system is consistent, write the solution. D. Identify the lead variables and free variables of a system represented by an augmented matrix in reduced row echelon form. E. Solve systems of linear equations using Gaussian elimination and back substitution. F. Solve systems of linear equations using Gauss-Jordan elimination. G. Model and solve application problems using linear systems. Reading: Linear Algebra 1.2 8 Matrices and Matrix Operations (a) vector, row vector, and column vector (b) equal matrices (c) scalar multiplication (d) sum of matrices (e) zero matrix (f) scalar product (g) linear combination (h) matrix multiplication (i) transpose (j) trace (k) identity matrix B. Perform the operations of matrix addition, scalar multiplication, transposition, trace, and matrix multiplication. C. Represent matrices in terms of double subscript notation. Reading: Linear Algebra 1.3 9 Inverses; Rules of Matrix Arithmetic (a) commutative property (b) singular (c) nonsingular or invertible (d) multiplicative inverse B. Recall, demonstrate, and apply algebraic properties for matrices. C. Recall that matrix multiplication is not commutative in general. Determine conditions under which matrices do commute. D. Recall and prove properties and identities involving the transpose operator. E. Recall and prove properties and identities involving matrix inverses. F. Recall and prove properties and identities involving matrix powers. G. Recall, demonstrate, and apply that the cancelation laws for scalar multiplication do not hold for matrix multiplication. H. Recall and apply the formula for the inverse of 2 2 matrices. Reading: Linear Algebra 1.4

10 Elementary Matrices (a) elementary matrix (b) row equivalent matrices B. Identify elementary matrices and find their inverses or show that their inverse does not exist. C. Relate elementary matrices to row operations. D. Factor matrices using elementary matrices. E. Find the inverse of a matrix, if possible, using elementary matrices. F. Prove theorems about matrix products and matrix inverses. G. Solve a linear equation using matrix inverses. Reading: Linear Algebra 1.5 11 Further Results on Systems of Equations and Invertibility A. Solve matrix equations using matrix algebra. B. Recall and prove properties and identities involving matrix inverses. C. Recall equivalent conditions for invertibility. Reading: Linear Algebra 1.6 12 Further Results on Systems of Equations and Invertibility (a) diagonal matrix (b) upper and lower triangular matrices (c) symmetric matrix (d) skew-symmetric matrix B. Determine powers of diagonal matrices. C. Recall and prove properties and identities involving the transpose operator. D. Prove basic facts involving symmetric and skew-symmetric matrices. Reading: Linear Algebra 1.7 13 Determinants (a) minor (b) cofactor (c) cofactor expansion (d) determinant (e) adjoint (f) Cramer s Rule B. Apply cofactor expansion to evaluate determinants of n n matrices. C. Recall and apply the properties of determinants to evaluate determinants. D. Evaluate the adjoint of a matrix. E. Determine whether or not a matrix has an inverse based on its determinant. F. Evaluate the inverse of a matrix using the adjoint method. G. Use Cramer s rule to solve a linear system. Reading: Linear Algebra 2.1

14 Properties of Determinants A. Recall the effects that row operations have on the determinants of matrices. Relate to the determinants of elementary matrices. B. Recall, apply and verify the properties of determinants to evaluate determinants, including: i. det(ab) = det(a) det(b) ii. det(ka) = k n det(a) iii. det(a 1 1 ) = det(a) iv. det(a T ) = det(a) iˇ. det(a) = 0 if and only if A is singular C. Evaluate the determinant of a matrix using row operations. D. Apply determinants to determine invertibility of matrix products. Reading: Linear Algebra 2.2,3 15 Euclidean n-space; Linear Transformations: Definitions and Examples A. Simplify, solve, and verify vector and matrix equations. B. Recall and compute the norm of a vector. C. Compute the inner product of two vectors. D. Recall and apply properties of the inner product. E. Determine if two vectors are orthogonal or not. F. Recall and apply the Cauchy-Schwarz Inequality. G. Define the following: (a) linear transformation (b) image (c) range H. Describe geometrically the effects of a linear operator. I. Determine whether or not a given transformation is linear. J. Prove theorems and solve application problems involving linear transformations. Reading: Linear Algebra 4.1,2 16 Properties of Linear Transformations Outcomes (a) standard matrix representation (b) eigenvalues and eigenvectors B. Determine the matrix that represents a given linear transformation of vectors given an algebraic description. C. Determine the matrix that represents a given linear transformation of vectors given a geometric description. D. Prove theorems and solve application problems involving linear transformations. Reading: Linear Algebra 4.3

17 Vector Spaces: Definitions and Examples (a) vector space (b) vector space axioms (c) vector space R n (d) vector space R m n (e) vector space of real-valued functions (f) additional properties of vector spaces B. Prove or disprove that a given set of vectors together with an addition and a scalar multiplication is a vector space. C. Prove and verify properties of a vector space. Reading: Linear Algebra 5.1 18 Subspaces (a) subspace (b) closure under addition (c) closure under scalar multiplication (d) zero subspace (e) linear combination (f) span (or subspace spanned by a set of vectors) (g) spanning set B. Prove or disprove that a set of vectors forms a subspace. C. Prove or disprove a set of vectors is a spanning set for R n. D. Prove or disprove a given vector is in the span of a set of vectors. Determine the span of a set of vectors. E. Prove theorems about vector spaces and spans. Reading: Linear Algebra 5.2 19 Linear Independence (a) linearly independent (b) linearly dependent (c) Wronskian B. Determine whether a set of vectors is linearly dependent or linearly independent. C. Geometrically describe the span of a set of vectors. For sets that are linearly dependent, determine a dependence relation. D. Prove theorems about linear independence. Reading: Linear Algebra 5.3

20 Basis and Dimension (a) basis (b) dimension (c) finite and infinite dimensional (d) standard basis B. Prove or disprove a set of vectors forms a basis. C. Find a basis for a vector space. D. Determine the dimension of a vector space. E. Geometrically interpret the ideas of span, linear dependance, basis, and dimension. Reading: Linear Algebra 5.4 21 Row Space, Column Space, and Null Space (a) row space (b) column space (c) null space (d) particular solution (e) general solution B. Express a product Ax as a linear combination of column vectors. C. Find a basis for a the column space, the row space, and the null space of a matrix. D. Find the basis for a span of vectors. Reading: Linear Algebra 5.5 22 Rank and Nullity (a) rank (b) nullity (c) The Consistency Theorem (d) equivalent statements of invertibility B. Find the rank and nullity of a matrix. C. Recall and prove identities involving rank and nullity D. Recall and apply the Consistency Theorm E. Recall and apply the equivalent statements of invertibility. Reading: Linear Algebra 5.6 23 Eigenvalues and Eigenvectors (a) eigenvalue or characteristic value (b) eigenvector or characteristic vector (c) characteristic polynomial or characteristic polynomial (d) equivalent statements of invertibility B. Find the eigenvalues and eigenvectors of an n n matrix. C. Prove theorems and solve application problems involving eigenvalues and eigenvectors. Reading: Linear Algebra 7.1

24 Diagonalization (a) diagonalizable (b) algebraic multiplicity (c) geometric multiplicity B. Determine whether or not a matrix is diagonalizable. C. Find the diagonalization of a matrix, if possible. D. Find powers of a matrix using the diagonalization of a matrix. E. Prove theorems and solve application problems involving the diagonalization of matrices. Reading: Linear Algebra 7.2 25 Limit, Continuity, Vector Derivative; The Rules of Differentiation (a) scalar functions (b) vector functions (c) components of a vector function (d) plane curve or space curve (e) parametrization of a curve (f) limit of a vector function (g) a vector function continuous at a point (h) derivative of a vector function (i) a differentiable vector function (j) integral of a vector function B. Graph a parametric curve. C. Identify a curve given its parametrization. D. Determine combinations of vector functions such as sums, vector products and scalar products. E. Evaluate limits, derivatives, and integrals of vector functions. F. Recall, derive and apply rules to combinations of vector functions for the following: (a) limits (b) differentiation (c) integration G. Determine continuity of a vector-valued function. H. Prove theorems involving limits and derivatives of vector-valued functions. I. Solve application problems involving vector-valued functions. Reading: Multivariable Calculus 14.1-2 26 Curves; Vector Calculus in Mechanics (a) directed path (b) differentiable parameterized curve (c) tangent vector (d) tangent line (e) unit tangent vector (f) principal normal vector (g) normal line (h) osculation plane (i) force vector (j) momentum vector (k) angular momentum vector (l) torque

B. Find the tangent vector and tangent line to a curve at a given point. C. Find the principle normal and normal line to a curve at a given point. D. Determine the osculating plane for a space curve at a given point. E. Reverse the direction of a curve. F. Solve application problems involving curves. G. Solve application problems involving force, momentum, angular momentum, and torque. Reading: Multivariable Calculus 14.3,6 Learning Module: Moving Trihedron 27 Arc Length (a) arc length (b) arc length parametrization B. Evaluate the arc length of a curve. C. Determine whether a curve is arc length parameterized. D. Find the arc length parametrization of a curve. Reading: Multivariable Calculus 14.4 28 Curvilinear Motion; Curvature (a) velocity vector function (b) speed (c) acceleration vector function (d) uniform circular motion (e) curvature (f) tangential component of acceleration (g) normal component of acceleration B. Given the position vector function of a moving object, calculate the velocity vector function, speed, and acceleration vector function, and vice versa. C. Calculate the curvature of a space curve. D. 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. E. Determine the tangential and normal components of acceleration for a given parameterized curve. F. Solve application problems involving curvilinear motion and curvature. Reading: Multivariable Calculus 14.5 29 Functions of Several Variables; A Brief Catalogue of the Quadric Surfaces; Projections (a) real-valued function of several variables (b) domain (c) range (d) bounded functions (e) quadric surface (f) intercepts (g) traces (h) sections (i) center

(j) symmetry (k) boundedness (l) cylinder (m) ellipsiod (n) elliptic cone (o) elliptic paraboloid (p) hyperboloid of one sheet (q) hyperboloid of two sheets (r) hyperbolic paraboloid (s) parabolic cylinder (t) elliptic cylinder (u) projection of a curve onto a coordinate plane B. Describe the domain and range of a function of several variables. C. Write a function of several variables given a description. D. Identify standard quadratic surfaces given their functions or graphs. E. Sketch the graph of a quadratic surface by sketching intercepts, traces, sections, centers, symmetry, boundedness. F. Find the projection of a curve, that is the intersection of two surfaces, to a coordinate plane. Reading: Multivariable Calculus 15.1-2 30 Graphs; Level Curves and Level Surfaces (a) level curve (b) level surface B. Describe the level sets of a function of several variables. C. Graphically represent a function of two variables by level curves or a function of three variables by level surfaces. D. Identify the characteristics of a function from its graph or from a graph of its level curves (or level surfaces). E. Solve application problems involving level sets. functions. Reading: Multivariable Calculus 15.3 31 Partial Derivatives (a) partial derivative of a function of several variables (b) second partial derivative (c) mixed partial derivative B. Interpret the definition of a partial derivative of a function of two variables graphically. C. Evaluate the partial derivatives of a function of several variables. D. Evaluate the higher order partial derivatives of a function of several variables. E. Verify equations involving partial derivatives. F. Apply partial derivatives to solve application problems. Reading: Multivariable Calculus 15.4 32 Open and Closed Sets; Limits and Continuity; Equity of Mixed Partials (a) neighborhood of a point (b) deleted neighborhood of a point (c) interior of a set (d) boundary of a set

(e) open set (f) closed set (g) limit of a function of several variables at a point (h) continuity of a function of several variables at a point B. Determine the boundary and interior of a set. C. Determine whether a set is open, closed, neither, or both. D. Evaluate the limit of a function of several variables or show that it does not exists. E. Determine whether or not a function is continuous at a given point. F. Recall and apply the conditions under which mixed partial derivatives are equal. Reading: Multivariable Calculus 15.5-6 33 Differentiability and Gradient (a) differentiable multivariable function (b) gradient of a multivariable function B. Evaluate the gradient of a function. C. Find a function with a given gradient. Reading: Multivariable Calculus 16.1 34 Gradient and Directional Derivative (a) directional derivative (b) isothermals B. Recall and prove identities involving gradients. C. Give a graphical interpretation of the gradient. D. Evaluate the directional derivative of a function. E. Give a graphical interpretation of directional derivative. F. 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 is then f( x) ) and it decreases most rapidly in the opposite direction (the rate of change is then f( x) ). G. Find the path of a heat seeking or a heat repelling particle. H. Solve application problems involving gradient and directional derivatives. Reading: Multivariable Calculus 16.2 35 The Mean-Value Theorem; The Chain Rule (a) the Mean Value Theorem for functions of several variables (b) normal line (c) chain rules for functions of several variables (d) implicit differentiation B. Recall and apply the Mean Value Theorem for functions of several variables and its corollaries. C. Apply an appropriate chain rule to evaluate a rate of change. D. Apply implicit differentiation to evaluate rates of change. E. Solve application problems involving chain rules and implicit differentiation. Reading: Multivariable Calculus 16.3

36 The Gradient as a Normal; Tangent Lines and Tangent Planes (a) normal vector (b) tangent vector (c) tangent line (d) tangent plane (e) normal line B. Use gradients to find the normal vector and normal line to a smooth planar curve at a given point. C. Use gradients to find the tangent vector and tangent line to a smooth planar curve at a given point. D. Use gradients to find the normal vector to a smooth surface at a given point. E. Use gradients to find the tangent plane to a smooth surface at a given point. F. Use gradients to find the normal line to a smooth surface at a given point. G. Solve application problems involving normals and tangents to curves and surfaces. Reading: Multivariable Calculus 16.4 37 Local Extreme Values (a) local minimum and local maximum (b) critical points (c) stationary points (d) saddle points (e) discriminant (f) Second Derivative Test B. Find the critical points of a function of two variables. C. Apply the Second-Partials Test to determine whether each critical point is a local minimum, a local maximum, or a saddle point. D. Solve word problems involving local extreme values. Reading: Multivariable Calculus 16.5 38 Absolute Extreme Values (a) absolute minimum and absolute maximum (b) bounded subset of a plane or three-space (c) the Extreme Value Theorem B. Determine absolute extreme values of a function defined on a closed and bounded set. C. Apply the Extreme Value Theorem to justify the method for finding extreme values of functions defined on certain sets. D. Solve word problems involving absolute extreme values. Reading: Multivariable Calculus 16.6 39 Maxima and Minima with Side Conditions (a) side conditions or constraints (b) method of Lagrange (c) Lagrange multipliers (d) cross-product equation of the Lagrange condition B. Graphically interpret the method of Lagrange. C. Determine the extreme values of a function subject to a side conditions by applying the method of Lagrange.

D. Apply the cross-product equation of the Lagrange condition to solve extreme value problems subject to side conditions. E. Apply the method of Lagrange to solve word problems. Reading: Multivariable Calculus 16.7 40 Differentials; Reconstructing a Function from its Gradient (a) differential (b) general solution (c) particular solution (d) connected open set (e) open region (f) simple closed curve (g) simply connected open region (h) partial derivative gradient test B. Determine the differential for a given function of several variables. C. Determine whether or not a vector function is a gradient. D. Given a vector function that is a gradient, find the functions with that gradient. Reading: Multivariable Calculus 16.8,9 41 Multiple-Sigma Notation; The Double Integral over a Rectangle R; The Evaluation of Double Integrals by Repeated Integrals (a) double sigma notation (b) triple sigma notation (c) upper sum (d) lower sum (e) double integral (f) integral formula for the volume of a solid bounded between a region Ω in the xy-plane and the graph of a non-negative function z = f(x, y) defined on Ω. (g) integral formula for the area of region in a plane (h) integral formula for the average of a function defined on a region Ω. (i) projection of a region onto a coordinate axis (j) Type I and Type II regions (k) reduction formulas for double integrals (l) the geometric interpretation of the reduction formulas for double integrals B. Evaluate double and triple sums given their sigma notation. C. Recall and apply summation identities. D. Approximate the integral of a function by a lower sum and an upper sum. E. Evaluate the integral of a function using the definition. F. Evaluate double integrals over a rectangle using the reduction formulas. G. Sketch planar regions and determine if they are Type I, Type II, or both. H. Evaluate double integrals over Type I and Type II regions. I. Change the order of integration of an integral. J. Apply double integrals to calculate volumes, areas, and averages. Reading: Multivariable Calculus 17.1,2,3

42 The Double Integral as the Limit of Riemann Sums; Polar Coordinates (a) diameter of a set (b) Riemann sum (c) double integral as a limit of Riemann sums (d) polar coordinates (r, θ) (e) transformation formulas between Cartesian and polar coordinates (f) double integral conversion formula between Cartesian and polar coordinates B. Represent a region in both Cartesian and polar coordinates. C. Evaluate double integrals in terms of polar coordinates. D. Evaluate areas and volumes using polar coordinates. E. Convert a double integral in Cartesian coordinates to a double integral in polar coordinates and then evaluate. Reading: Multivariable Calculus 17.4 43 Further Applications of the Double Integral (a) integral formula for the mass of a plate (b) integral formulas for the center of mass of a plate (c) integral formulas for the centroid of a plate (d) integral formulas for the moment of an inertia of a plate (e) radius of gyration (f) the Parallel Axis Theorem B. Evaluate the mass and center or mass of a plate C. Evaluate the centroid of a plate. D. Evaluate the moments of inertia of a plate. E. Calculate the radius of gyration of a plate. F. Recall and apply the parallel axis theorem. Reading: Multivariable Calculus 17.5 44 Triple Integrals; Reduction to Repeated Integrals (a) triple integral (b) integral formula for the volume of a solid (c) integral formula for the mass of a solid (d) integral formulas for the center of mass of a solid B. Evaluate physical quantities using triple integrals such as volume, mass, center of mass, and moments of intertia. C. Recall and apply the properties of triple integrals, including: linearity, order, additivity, and the mean-value condition. D. Sketch the domain of integration of an iterated integral. E. Change the order of integration of a triple integral. Reading: Multivariable Calculus 17.6-7

45 Cylindrical Coordinates (a) cylindrical coordinates of a point (b) coordinate transformations between Cartesian and cylindrical coordinates (c) cylindrical element of volume B. Convert between Cartesian and cylindrical coordinates. C. Describe regions in cylindrical coordinates. D. Evaluate triple integrals using cylindrical coordinates. Reading: Multivariable Calculus 17.8 46 Spherical Coordinates (a) spherical coordinates of a point (b) coordinate transformations between Cartesian and spherical coordinates (c) spherical element of volume B. Convert between Cartesian and spherical coordinates. C. Describe regions in spherical coordinates. D. Evaluate triple integrals using spherical coordinates. Reading: Multivariable Calculus 17.9 47 Jacobians; Changing Variables in Multiple Integration (a) Jacobian (b) change of variable formula for double integration (c) change of variable formula for triple integration B. Find the Jacobian of a coordinate transformation. C. Use a coordinate transformation to evaluate double and triple integrals. Reading: Multivariable Calculus 17.10 48 Line Integrals (a) work along a curved path (b) smooth parametric curve (c) directed or oriented curve (d) path dependence (e) closed curve B. Evaluate the work done by a varying force over a curved path. C. Evaluate line integrals in general including line integrals with respect to arc length. D. Evaluate the physical characteristics of a wire such as centroid, mass, and center of mass using line integrals. E. Determine whether or not a vector field is a gradient. F. Determine whether or not a differential form is exact. Reading: Multivariable Calculus 18.1,4

49 The Fundamental Theorem for Line Integrals; Work-Energy Formula; Conservation of Mechanical Energy (a) path-independent line integrals (b) closed vector field (c) simply connected B. Recall, apply, and verify the Fundamental Theorem for Line Integrals (Theorem 18.2.1 in Section 18.2). C. Determine whether or not a force field is closed on a given region, and if so, find its potential function. D. Solve application problems involving work done by a conservative vector field Reading: Multivariable Calculus 18.2,3 50 Green s Theorem (a) Jordan curve (b) Jordan region (c) Green s Theorem B. Recall and verify Green s Theorem. C. Apply Green s Theorem to evaluate line integrals. D. Apply Green s Theorem to find the area of a region. E. Derive identities involving Green s Theorem Reading: Multivariable Calculus 18.5 51 Parameterized Surfaces; Surface Area (a) parameterized surface (b) fundamental vector product (c) element of surface area for a parameterized surface (d) surface integral (e) integral formula for the surface area of a parameterized surface (f) integral formula for the surface area of a surface z = f(x, y) (g) upward unit normal B. parameterize a surface. C. evaluate the fundamental vector product for a parameterized surface. D. Calculate the surface area of a parameterized surface. E. Calculate the surface area of a surface z = f(x, y). Reading: Multivariable Calculus 18.6 52 Surface Integrals (a) surface integral (b) integral formulas for the surface area and centroid of a parameterized surface (c) integral formulas for the mass and center of mass of a parameterized surface (d) integral formulas for the moments of inertia of a parameterized surface (e) integral formula for flux through a surface B. Calculate the surface area and centroid of a parameterized surface. C. Calculate the mass and center of mass of a parameterized surface. D. Calculate the moments of inertia of a parameterized surface. E. Evaluate the flux of a vector field through a surface. F. Solve application problems involving surface integrals. Reading: Multivariable Calculus 18.7

53 The Vector Differential Operator (a) the vector differential operator (b) divergence (c) curl (d) Laplacian B. Evaluate the divergence of a vector field. C. Evaluate the curl of a vector field D. Evaluate the Laplacian of a function. E. Recall, derive and apply formulas involving divergence, gradient and Laplacian. F. Interpret that divergence and curl of a vector fields physically. Reading: Multivariable Calculus 18.8 54 The Divergence Theorem (a) outward unit normal (b) the divergence theorem (c) sink and source (d) solenoidal B. Recall and verify the Divergence Theorem. C. Apply the Divergence Theorem to evaluate the flux through a surface. D. Solve application problems using the Divergence Theorem. Reading: Multivariable Calculus 18.9 55 Stokes Theorem (a) oriented surface (b) outward, upward, and downward unit normal (c) the positive sense around the boundary of a surface (d) circulation (e) component of curl in the normal direction (f) irrotational (g) Stokes theorem B. Recall and verify Stoke s theorem. C. Use Stokes Theorem to calculate the flux of a curl vector field through a surface by a line integral. D. Apply Stoke s theorem to calculate the work (or circulation) of a vector field around a simple closed curve. Reading: Multivariable Calculus 18.10