Math 302 Outcome Statements Winter 2013
|
|
- Moses Anthony
- 5 years ago
- Views:
Transcription
1 Math 302 Outcome Statements Winter 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 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
2 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 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
3 (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 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 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 Gaussian elimination (a) reduced row echelon form (b) leading variables or pivots (c) free variables
4 (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 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 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
5 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 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 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 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
6 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
7 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 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 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
8 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 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 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 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
9 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 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 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
10 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 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 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
11 (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 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 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 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
12 (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 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 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 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
13 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 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 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 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.
14 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 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
15 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 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 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
16 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 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 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 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
17 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 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 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 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
18 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 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 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
Upon successful completion of MATH 220, the student will be able to:
MATH 220 Matrices Upon successful completion of MATH 220, the student will be able to: 1. Identify a system of linear equations (or linear system) and describe its solution set 2. Write down the coefficient
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationLinear Algebra I for Science (NYC)
Element No. 1: To express concrete problems as linear equations. To solve systems of linear equations using matrices. Topic: MATRICES 1.1 Give the definition of a matrix, identify the elements and the
More informationMATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am-12:00pm.
MATH 2083 FINAL EXAM REVIEW The final exam will be on Wednesday, May 4 from 10:00am-12:00pm. Bring a calculator and something to write with. Also, you will be allowed to bring in one 8.5 11 sheet of paper
More informationhomogeneous 71 hyperplane 10 hyperplane 34 hyperplane 69 identity map 171 identity map 186 identity map 206 identity matrix 110 identity matrix 45
address 12 adjoint matrix 118 alternating 112 alternating 203 angle 159 angle 33 angle 60 area 120 associative 180 augmented matrix 11 axes 5 Axiom of Choice 153 basis 178 basis 210 basis 74 basis test
More informationMULTIVARIABLE CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL EQUATIONS
T H I R D E D I T I O N MULTIVARIABLE CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL EQUATIONS STANLEY I. GROSSMAN University of Montana and University College London SAUNDERS COLLEGE PUBLISHING HARCOURT BRACE
More informationMATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT
MATHEMATICS COMPREHENSIVE EXAM: IN-CLASS COMPONENT The following is the list of questions for the oral exam. At the same time, these questions represent all topics for the written exam. The procedure for
More information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
More informationThe value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver I.N.
Math 410 Homework Problems In the following pages you will find all of the homework problems for the semester. Homework should be written out neatly and stapled and turned in at the beginning of class
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More informationSUMMARY OF MATH 1600
SUMMARY OF MATH 1600 Note: The following list is intended as a study guide for the final exam. It is a continuation of the study guide for the midterm. It does not claim to be a comprehensive list. You
More informationMath 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationMath 1553, Introduction to Linear Algebra
Learning goals articulate what students are expected to be able to do in a course that can be measured. This course has course-level learning goals that pertain to the entire course, and section-level
More informationGlossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the
More informationEquality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.
Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More information2. TRIGONOMETRY 3. COORDINATEGEOMETRY: TWO DIMENSIONS
1 TEACHERS RECRUITMENT BOARD, TRIPURA (TRBT) EDUCATION (SCHOOL) DEPARTMENT, GOVT. OF TRIPURA SYLLABUS: MATHEMATICS (MCQs OF 150 MARKS) SELECTION TEST FOR POST GRADUATE TEACHER(STPGT): 2016 1. ALGEBRA Sets:
More informationCalculus Early Transcendentals
Calculus Early Transcendentals 978-1-63545-101-6 To learn more about all our offerings Visit Knewton.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Gilbert Strang, Massachusetts
More information1 Geometry of R Conic Sections Parametric Equations More Parametric Equations Polar Coordinates...
Contents 1 Geometry of R 2 2 1.1 Conic Sections............................................ 2 1.2 Parametric Equations........................................ 3 1.3 More Parametric Equations.....................................
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More information1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?
. Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More informationLAKELAND COMMUNITY COLLEGE COURSE OUTLINE FORM
LAKELAND COMMUNITY COLLEGE COURSE OUTLINE FORM ORIGINATION DATE: 8/2/99 APPROVAL DATE: 3/22/12 LAST MODIFICATION DATE: 3/28/12 EFFECTIVE TERM/YEAR: FALL/ 12 COURSE ID: COURSE TITLE: MATH2800 Linear Algebra
More informationCURRENT MATERIAL: Vector Calculus.
Math 275, section 002 (Ultman) Spring 2012 FINAL EXAM REVIEW The final exam will be held on Wednesday 9 May from 8:00 10:00am in our regular classroom. You will be allowed both sides of two 8.5 11 sheets
More informationExtra Problems for Math 2050 Linear Algebra I
Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as
More information1 9/5 Matrices, vectors, and their applications
1 9/5 Matrices, vectors, and their applications Algebra: study of objects and operations on them. Linear algebra: object: matrices and vectors. operations: addition, multiplication etc. Algorithms/Geometric
More informationMath Linear Algebra Final Exam Review Sheet
Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationMath 118, Fall 2014 Final Exam
Math 8, Fall 4 Final Exam True or false Please circle your choice; no explanation is necessary True There is a linear transformation T such that T e ) = e and T e ) = e Solution Since T is linear, if T
More informationDetailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors
Unit 1 Vectors In this unit, we introduce vectors, vector operations, and equations of lines and planes. Note: Unit 1 is based on Chapter 12 of the textbook, Salas and Hille s Calculus: Several Variables,
More informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
More informationCourse Outline. 2. Vectors in V 3.
1. Vectors in V 2. Course Outline a. Vectors and scalars. The magnitude and direction of a vector. The zero vector. b. Graphical vector algebra. c. Vectors in component form. Vector algebra with components.
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationMATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION
MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether
More informationMATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics
MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones
More informationLinear Algebra Highlights
Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to
More informationMAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to:
MAC 2 Module Determinants Learning Objectives Upon completing this module, you should be able to:. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor
More informationMATH 332: Vector Analysis Summer 2005 Homework
MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,
More informationConceptual Questions for Review
Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.
More informationMATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.
MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of
More informationLinear Algebra Homework and Study Guide
Linear Algebra Homework and Study Guide Phil R. Smith, Ph.D. February 28, 20 Homework Problem Sets Organized by Learning Outcomes Test I: Systems of Linear Equations; Matrices Lesson. Give examples of
More informationMultivariable Calculus, Applications and Theory. Kenneth Kuttler
Multivariable Calculus, Applications and Theory Kenneth Kuttler August 19, 211 2 Contents.1 Introduction.................................. 8 I Basic Linear Algebra 11 1 Fundamentals 13 1..1 Outcomes...............................
More informationMATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003
MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003 1. True or False (28 points, 2 each) T or F If V is a vector space
More informationWe wish the reader success in future encounters with the concepts of linear algebra.
Afterword Our path through linear algebra has emphasized spaces of vectors in dimension 2, 3, and 4 as a means of introducing concepts which go forward to IRn for arbitrary n. But linear algebra does not
More informationA Brief Outline of Math 355
A Brief Outline of Math 355 Lecture 1 The geometry of linear equations; elimination with matrices A system of m linear equations with n unknowns can be thought of geometrically as m hyperplanes intersecting
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationChapter 5 Eigenvalues and Eigenvectors
Chapter 5 Eigenvalues and Eigenvectors Outline 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 2 5.1 Eigenvalues and Eigenvectors Eigenvalue and Eigenvector If A is a n n
More informationAPPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF
ELEMENTARY LINEAR ALGEBRA WORKBOOK/FOR USE WITH RON LARSON S TEXTBOOK ELEMENTARY LINEAR ALGEBRA CREATED BY SHANNON MARTIN MYERS APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF When you are done
More informationChapter 7. Linear Algebra: Matrices, Vectors,
Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.
More informationHarbor Creek School District
Unit 1 Days 1-9 Evaluate one-sided two-sided limits, given the graph of a function. Limits, Evaluate limits using tables calculators. Continuity Evaluate limits using direct substitution. Differentiability
More information1 Geometry of R Conic Sections Parametric Equations More Parametric Equations Polar Coordinates...
Contents 1 Geometry of R 1.1 Conic Sections............................................ 1. Parametric Equations........................................ 3 1.3 More Parametric Equations.....................................
More informationDepartment: Course Description: Course Competencies: MAT 201 Calculus III Prerequisite: MAT Credit Hours (Lecture) Mathematics
Department: Mathematics Course Description: Calculus III is the final course in the three-semester sequence of calculus courses. This course is designed to prepare students to be successful in Differential
More informationMAT188H1S LINEAR ALGEBRA: Course Information as of February 2, Calendar Description:
MAT188H1S LINEAR ALGEBRA: Course Information as of February 2, 2019 2018-2019 Calendar Description: This course covers systems of linear equations and Gaussian elimination, applications; vectors in R n,
More informationMobile Robotics 1. A Compact Course on Linear Algebra. Giorgio Grisetti
Mobile Robotics 1 A Compact Course on Linear Algebra Giorgio Grisetti SA-1 Vectors Arrays of numbers They represent a point in a n dimensional space 2 Vectors: Scalar Product Scalar-Vector Product Changes
More informationANSWERS (5 points) Let A be a 2 2 matrix such that A =. Compute A. 2
MATH 7- Final Exam Sample Problems Spring 7 ANSWERS ) ) ). 5 points) Let A be a matrix such that A =. Compute A. ) A = A ) = ) = ). 5 points) State ) the definition of norm, ) the Cauchy-Schwartz inequality
More informationOHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1
OHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1 (37) If a bug walks on the sphere x 2 + y 2 + z 2 + 2x 2y 4z 3 = 0 how close and how far can it get from the origin? Solution: Complete
More informationMathematics Syllabus UNIT I ALGEBRA : 1. SETS, RELATIONS AND FUNCTIONS
Mathematics Syllabus UNIT I ALGEBRA : 1. SETS, RELATIONS AND FUNCTIONS (i) Sets and their Representations: Finite and Infinite sets; Empty set; Equal sets; Subsets; Power set; Universal set; Venn Diagrams;
More informationMATHEMATICS. Units Topics Marks I Relations and Functions 10
MATHEMATICS Course Structure Units Topics Marks I Relations and Functions 10 II Algebra 13 III Calculus 44 IV Vectors and 3-D Geometry 17 V Linear Programming 6 VI Probability 10 Total 100 Course Syllabus
More informationMAT Linear Algebra Collection of sample exams
MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
More informationa s 1.3 Matrix Multiplication. Know how to multiply two matrices and be able to write down the formula
Syllabus for Math 308, Paul Smith Book: Kolman-Hill Chapter 1. Linear Equations and Matrices 1.1 Systems of Linear Equations Definition of a linear equation and a solution to a linear equations. Meaning
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More informationLinear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
More informationCURRENT MATERIAL: Vector Calculus.
Math 275, section 002 (Ultman) Fall 2011 FINAL EXAM REVIEW The final exam will be held on Wednesday 14 December from 10:30am 12:30pm in our regular classroom. You will be allowed both sides of an 8.5 11
More informationAssignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work.
Assignment 1 Math 5341 Linear Algebra Review Give complete answers to each of the following questions Show all of your work Note: You might struggle with some of these questions, either because it has
More informationENGINEERING MATHEMATICS I. CODE: 10 MAT 11 IA Marks: 25 Hrs/Week: 04 Exam Hrs: 03 PART-A
ENGINEERING MATHEMATICS I CODE: 10 MAT 11 IA Marks: 25 Hrs/Week: 04 Exam Hrs: 03 Total Hrs: 52 Exam Marks:100 PART-A Unit-I: DIFFERENTIAL CALCULUS - 1 Determination of n th derivative of standard functions-illustrative
More informationSolving a system by back-substitution, checking consistency of a system (no rows of the form
MATH 520 LEARNING OBJECTIVES SPRING 2017 BROWN UNIVERSITY SAMUEL S. WATSON Week 1 (23 Jan through 27 Jan) Definition of a system of linear equations, definition of a solution of a linear system, elementary
More informationDot Products. K. Behrend. April 3, Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem.
Dot Products K. Behrend April 3, 008 Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem. Contents The dot product 3. Length of a vector........................
More informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
More informationMath 307 Learning Goals
Math 307 Learning Goals May 14, 2018 Chapter 1 Linear Equations 1.1 Solving Linear Equations Write a system of linear equations using matrix notation. Use Gaussian elimination to bring a system of linear
More informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.
More informationFall 2016 MATH*1160 Final Exam
Fall 2016 MATH*1160 Final Exam Last name: (PRINT) First name: Student #: Instructor: M. R. Garvie Dec 16, 2016 INSTRUCTIONS: 1. The exam is 2 hours long. Do NOT start until instructed. You may use blank
More informationSTATE COUNCIL OF EDUCATIONAL RESEARCH AND TRAINING TNCF DRAFT SYLLABUS.
STATE COUNCIL OF EDUCATIONAL RESEARCH AND TRAINING TNCF 2017 - DRAFT SYLLABUS Subject :Mathematics Class : XI TOPIC CONTENT Unit 1 : Real Numbers - Revision : Rational, Irrational Numbers, Basic Algebra
More information1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4
MATH2202 Notebook 1 Fall 2015/2016 prepared by Professor Jenny Baglivo Contents 1 MATH2202 Notebook 1 3 1.1 Single Variable Calculus versus Multivariable Calculus................... 3 1.2 Rectangular Coordinate
More informationIntroduction to Linear Algebra, Second Edition, Serge Lange
Introduction to Linear Algebra, Second Edition, Serge Lange Chapter I: Vectors R n defined. Addition and scalar multiplication in R n. Two geometric interpretations for a vector: point and displacement.
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationexample consider flow of water in a pipe. At each point in the pipe, the water molecule has a velocity
Module 1: A Crash Course in Vectors Lecture 1: Scalar and Vector Fields Objectives In this lecture you will learn the following Learn about the concept of field Know the difference between a scalar field
More informationCourse Notes Math 275 Boise State University. Shari Ultman
Course Notes Math 275 Boise State University Shari Ultman Fall 2017 Contents 1 Vectors 1 1.1 Introduction to 3-Space & Vectors.............. 3 1.2 Working With Vectors.................... 7 1.3 Introduction
More informationMATH 1020 WORKSHEET 12.1 & 12.2 Vectors in the Plane
MATH 100 WORKSHEET 1.1 & 1. Vectors in the Plane Find the vector v where u =, 1 and w = 1, given the equation v = u w. Solution. v = u w =, 1 1, =, 1 +, 4 =, 1 4 = 0, 5 Find the magnitude of v = 4, 3 Solution.
More informationMathTutor DVD South Africa 2011
Young Minds DVD - Learn to recognize the numbers 1-10. - Associate the number with physical objects. - Count from 1-10. - Animal names and sounds. - Fruits and Vegetables. - Names of colours. - Names of
More informationMultiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015
Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction
More informationMATH 2360 REVIEW PROBLEMS
MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) 1 2 2 1 (a) 0 1 1 2 ( ) 0 1 2 (b) 0 3 1 4 3 4 5 2 5 (c) 0 3 ) 1 4 ( 1
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More informationSOLUTIONS TO THE FINAL EXAM. December 14, 2010, 9:00am-12:00 (3 hours)
SOLUTIONS TO THE 18.02 FINAL EXAM BJORN POONEN December 14, 2010, 9:00am-12:00 (3 hours) 1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 2/13/13, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.2. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241sp13/241.html)
More informationIndex. Excerpt from "Precalculus" 2014 AoPS Inc. Copyrighted Material INDEX
Index, 29 \, 6, 29 \, 6 [, 5 1, 2!, 29?, 309, 179, 179, 29 30-60-90 triangle, 32 45-45-90 triangle, 32 Agnesi, Maria Gaetana, 191 Airy, Sir George, 180 AMC, vii American Mathematics Competitions, see AMC
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationPRECALCULUS BISHOP KELLY HIGH SCHOOL BOISE, IDAHO. Prepared by Kristina L. Gazdik. March 2005
PRECALCULUS BISHOP KELLY HIGH SCHOOL BOISE, IDAHO Prepared by Kristina L. Gazdik March 2005 1 TABLE OF CONTENTS Course Description.3 Scope and Sequence 4 Content Outlines UNIT I: FUNCTIONS AND THEIR GRAPHS
More informationLinear Algebra M1 - FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8.
Linear Algebra M1 - FIB Contents: 5 Matrices, systems of linear equations and determinants 6 Vector space 7 Linear maps 8 Diagonalization Anna de Mier Montserrat Maureso Dept Matemàtica Aplicada II Translation:
More informationMULTIVARIABLE CALCULUS 61
MULTIVARIABLE CALCULUS 61 Description Multivariable Calculus is a rigorous second year course in college level calculus. This course provides an in-depth study of vectors and the calculus of several variables
More informationHands-on Matrix Algebra Using R
Preface vii 1. R Preliminaries 1 1.1 Matrix Defined, Deeper Understanding Using Software.. 1 1.2 Introduction, Why R?.................... 2 1.3 Obtaining R.......................... 4 1.4 Reference Manuals
More informationLINEAR SYSTEMS, MATRICES, AND VECTORS
ELEMENTARY LINEAR ALGEBRA WORKBOOK CREATED BY SHANNON MARTIN MYERS LINEAR SYSTEMS, MATRICES, AND VECTORS Now that I ve been teaching Linear Algebra for a few years, I thought it would be great to integrate
More informationIndex. Cambridge University Press Foundation Mathematics for the Physical Sciences K. F. Riley and M. P. Hobson.
absolute convergence of series, 225 acceleration vector, 449 addition rule for probabilities, 603, 608 adjoint, see Hermitian conjugate algebra of complex numbers, 177 8 matrices, 378 9 power series, 236
More informationDaily Update. Math 290: Elementary Linear Algebra Fall 2018
Daily Update Math 90: Elementary Linear Algebra Fall 08 Lecture 7: Tuesday, December 4 After reviewing the definitions of a linear transformation, and the kernel and range of a linear transformation, we
More informationLecture 23: 6.1 Inner Products
Lecture 23: 6.1 Inner Products Wei-Ta Chu 2008/12/17 Definition An inner product on a real vector space V is a function that associates a real number u, vwith each pair of vectors u and v in V in such
More informationCALC 3 CONCEPT PACKET Complete
CALC 3 CONCEPT PACKET Complete Written by Jeremy Robinson, Head Instructor Find Out More +Private Instruction +Review Sessions WWW.GRADEPEAK.COM Need Help? Online Private Instruction Anytime, Anywhere
More information