Multivariable Calculus (Part I) 13.1 Vectrs in Three-Dimensinal Space Math 302 Learning Objectives Plt pints in three-dimensinal space. Find the distance between tw pints in three-dimensinal space. Write the equatin f a sphere centered at a given pint with a given radius. Find symmetric pints abut: a pint, a line, r a plane. Parameterize a line segment. Find the midpint f a line segment. 13.2 Vectrs in Three-Dimensinal Space Define vectr. Represent a vectr in its crdinate frm, as a directed line segments and as a linear cmbinatin f unit vectrs. Perfrm basic peratins n vectrs including: additin, subtractin, and scalar multiplicatin. Interpret these peratins gemetrically. Find the nrm (magnitude r length) f a vectr. Knw and apply the nrm prperties f vectrs. Determine whether tw vectrs are parallel. Find the unit vectr in the directin f a given vectr. Explain the natural crrespndence between pints and vectrs. 13.3 The Dt Prduct Define dt prduct. Evaluate the dt prduct f tw vectrs. Knw and apply the prperties f dt prduct. Interpret the dt prduct gemetrically. Use the dt prduct t determine if tw vectrs are perpendicular (r rthgnal). Use the dt prduct t find the angle between tw vectrs. Find the prjectin and cmpnent f prjectin f ne vectr nt anther. Explain what these are gemetrically. State and prve the Schwartz inequality. 13.4 The Crss Prduct Define crss prduct. Evaluate the crss prduct f tw vectrs. Knw and apply the prperties f crss prduct.
Interpret the crss prduct gemetrically. Use the crss prduct t find a vectr perpendicular t tw given nnparallel vectrs. Use the crss prduct t find the area f a triangle r a parallelgram. Define scalar triple prduct. Use the scalar triple prduct t find the vlume f a parallelepiped. State and prve Lagrange s identity. 13.5 Lines Given infrmatin in a variety f ways, find the critical infrmatin needed t write the equatin f a line; namely, a pint and a directin vectr. Find the equatin f the line represented in any ne f the fllwing ways: Vectr parameterizatin Scalar parametric equatins Symmetric frm Determine whether a pair f lines is intersecting, parallel, r skew. Find the angle between tw lines. Find the distance frm a pint t a line. 13.6 Planes Given infrmatin in a variety f ways, find the critical infrmatin needed t write the equatin f a plane; namely, a pint and a nrmal vectr. Find the equatin f the plane represented in either f the fllwing ways: Scalar equatin Vectr equatin Symmetric frm Determine whether a cllectin f vectrs is cplanar. Find the distance frm a pint t a plane. 13.7 Higher dimensins Generalize the fllwing fr R n. The distance between tw pints. The midpint f a line segment. The sum, difference and scalar multiples f vectrs. Dt prduct. The nrm f a vectr. The angle between tw vectrs.
Linear Algebra 2.1 Systems f Linear Equatins Define linear equatin system f linear equatins slutin slutin set Identify linear equatins and systems f linear equatins Relate the fllwing types f slutin sets f a system f tw r three variables t the intersectins f lines in a plane r the intersectin f planes in three space: a unique slutin. infinitely many slutins. n slutin. Represent a linear system as an augmented matrix and vice versa. Transfrm a system t a triangular pattern and then apply back substitutin t slve the linear system. Represent the slutin set t a linear system using parametric equatins. 2.2 Direct Methds fr Slving Linear Systems Identify matrices that are in rw echeln frm and reduced rw echeln frm. Determine whether a system f linear equatins has n slutin, a unique slutin r an infinite number f slutins frm its echeln frm. Apply elementary rw peratins t transfrm systems f linear equatins. Slve systems f linear equatins using: Gaussian eliminatin Gauss-Jrdan eliminatin Define and evaluate the rank f a matrix. Apply the Rank Therem relate the rank f an augmented matrix t the slutin set f a system in the case f: hmgeneus systems nnhmgeneus systems Mdel and slve applicatin prblems using linear systems. 2.3 Spanning Sets and Linear Independence Explain what is meant by the span f a set f vectrs bth gemetrically and algebraically. Determine the span f a set f vectrs. Determine if a given vectr is in the span f a set f vectrs. Define linear independence. Determine whether a set f vectrs is linearly dependent r linearly independent.
Fr sets that are linearly dependent, determine a dependence relatin. Prve therems abut span and linear independence. 3.1 Matrix Operatins Perfrm the peratins f: matrix additin scalar multiplicatin transpsitin matrix multiplicatin Represent matrices in terms f duble subscript ntatin. Recall and demnstrate that the cancellatin laws fr scalar multiplicatin d nt hld fr matrix multiplicatin. Use matrices and matrix peratins t mdel applicatin prblems. Represent and cmpute matrix prducts in terms f blcks. 3.2 Matrix Algebra Recall and apply the algebraic prperties fr matrix additin scalar multiplicatin n matrices matrix multiplicatin transpsitin Prve algebraic prperties fr matrices. Apply the cncepts f span and linear independence t matrices. Recall that matrix multiplicatin is nt cmmutative in general. Determine cnditins under which matrices d cmmute. 3.3 The Inverse f a Matrix Define the inverse f a matrix. Recall the Fundamental Therem f Invertible Matrices and the prperties f invertible matrices. Prve therems invlving matrix inverses. Recall and apply the frmula fr the inverse f 2 2 matrices. Demnstrate the relatinship between elementary matrices and rw peratins. Determine inverses f elementary matrices. Cmpute the inverse f a matrix using the Gauss-Jrdan methd. Slve a linear equatin using matrix inverses. 3.5 Subspaces, Basis, Dimensin, and Rank Define subspace R n. Determine whether r nt a given set f vectrs frms a subspace f R n.
Define fr a given matrix: rw space clumn space null space Determine whether r nt a given vectr is in a specified rw space, clumn space r null space. Define fr a subspace: basis dimensin Given a subspace, determine its dimensin and a basis. Verify whether r nt a given set f vectrs is a basis fr the subspace. Define fr a matrix: rank nullity Determine the rank and nullity f a given matrix. Prve and recall therems invlving the rank, nullity, and invertibility f matrices. 3.6 Linear Transfrmatins Define linear transfrmatin. Determine whether r nt a given transfrmatin is linear. Determine the matrix that represents a given linear transfrmatin f vectrs. Prve and recall therems invlving linear transfrmatins. Find cmpsitins and inverses f linear transfrmatins. 4.1 Intrductin t Eigenvalues and Eigenvectrs Interpret the eigenvalue prblem: algebraically gemetrically Determine whether a given vectr is an eigenvectr. Verify that a given value is an eigenvalue. Determine eigenvalues and eigenvectrs: gemetrically frm the graph f the eigenspace Find the eigenvalues and eigenvectrs f a general 2 2 matrix. 4.2 Determinants Apply the Laplace Expansin t evaluate determinants f n n matrices. Recall and apply the prperties f determinants t evaluate determinants, including: det(ab) = det(a) det(b) det(ka) = k n det(a) det(a -1 )= 1/det(A)
det(a T )=det(a) Recall the effects that rw peratins have n the determinants f matrices. Relate elementary matrices t rw peratins. Prve identities invlving determinants. Evaluate matrix inverses using the adjint methd. Determine whether r nt a matrix has an inverse based n its determinant. Use Cramer's rule t slve a linear system. 4.3 Eigenvalues and Eigenvectrs f n n Matrices Given an n n matrix, cmpute the characteristic plynmial the Eigenvalues a basis fr each Eigenspace the algebraic and gemetric multiplicities f each Eigenvalue Slve applicatin prblems invlving Eigenvalues and Eigenvectrs. Recall and prve therems invlving Eigenvalues and Eigenvectrs. 4.4 Similarity and Diagnalizatin Define similar matrices. Determine whether r nt tw matrices are similar. Find the diagnalizatin f a matrix r determine that the matrix is nt diagnalizable. Find pwers f a matrix using the diagnalizatin f a matrix. Prve therems invlving the similarity and diagnalizatin f matrices. Multivariable Calculus (Part II) 14.1 Limit, Cntinuity, Vectr Derivative scalar functins vectr functins cmpnents f a vectr functin plane curve r space curve parametrizatin f a curve limit f a vectr functin a vectr functin cntinuus at a pint derivative f a vectr functin integral f a vectr functin Graph a parametric curve. Identify a curve given its parametrizatin. Determine cmbinatins f vectr functins such as sums, vectr prducts and scalar prducts.
Evaluate limits, derivatives, and integrals f vectr functins. Recall, derive and apply rules fr the fllwing: limits differentiatin integratin Determine cntinuity f a vectr-valued functin. 14.2 The Rules f Differentiatin Recall, derive and apply differentiatin rules t cmbinatins f vectr functins Prve therems invlving derivatives f vectr-valued functins. Slve applicatin prblems invlving derivatives f vectr-valued functins. 14.3 Curves directed path differentiable parameterized curve tangent vectr tangent line unit tangent vectr principal nrmal vectr nrmal line sculating plane Find the tangent vectr and tangent line t a curve at a given pint. Find the principle nrmal and nrmal line t a curve at a given pint. Determine the sculating plane fr a space curve at a given pint. Reverse the directin f a curve. Slve applicatin prblems invlving curves. 14.4 Arc Length arc length arc length parametrizatin Evaluate the arc length f a curve. Determine whether a curve is arc length parameterized. Find the arc length parametrizatin f a curve, if pssible. 14.5 Curvilinear Mtin; Curvature velcity vectr functin speed
acceleratin vectr functin unifrm circular mtin curvature tangential cmpnent f acceleratin nrmal cmpnent f acceleratin Given the psitin vectr functin f a mving bject, calculate the velcity vectr functin, speed, and acceleratin vectr functin, and vice versa. Calculate the curvature f a space curve. Recall the frmulas fr the curvature f a parameterized planar curve r a planar curve that is the graph f a functin. Apply these frmulas t calculate the curvature f a planar curve. Determine the tangential and nrmal cmpnents f acceleratin fr a given parameterized curve. Slve applicatin prblems invlving curvilinear mtin and curvature. 15.1 Elementary Examples real-valued functin f several variables dmain range bunded functins Describe the dmain and range f a functin f several variables. Write a functin f several variables given a descriptin. 15.2 A Brief Catalgue f the Quadric Surfaces; Prjectins quadric surface intercepts traces sectins symmetry bundedness cylinder ellipsid elliptic cne elliptic parablid hyperblid f ne sheet hyperblid f tw sheets hyperblic parablid parablic cylinder elliptic cylinder prjectin f a curve nt a crdinate plane
Identify standard quadratic surfaces given their functins r graphs. Sketch the graph f a quadratic surface by sketching intercepts, traces, sectins, centers, symmetry, bundedness. Find the prjectin f a curve, that is the intersectin f tw surfaces, t a crdinate plane. 15.3 Graphs; Level Curves and Level Surfaces level curve level surface Describe the level sets f a functin f several variables. Graphically represent a functin f tw variables by level curves r a functin f three variables by level surfaces. Identify the characteristics f a functin frm its graph r frm a graph f its level curves (r level surfaces). Slve applicatin prblems invlving level sets. 15.4 Partial Derivatives partial derivative f a functin f several variables secnd partial derivative mixed partial derivative Interpret the definitin f a partial derivative f a functin f tw variables graphically. Evaluate the partial derivatives f a functin f several variables. Evaluate the higher rder partial derivatives f a functin f several variables. Verify equatins invlving partial derivatives. Apply partial derivatives t slve applicatin prblems. 15.5 Open and Clsed Sets neighbrhd f a pint deleted neighbrhd f a pint interir f a set bundary f a set pen set clsed set Determine the bundary and interir f a set. Determine whether a set is pen, clsed, neither, r bth. 15.6 Limits and Cntinuity; Equality f Mixed Partials
limit f a functin f several variables at a pint cntinuity f a functin f several variables at a pint Evaluate the limit f a functin f several variables r shw that it des nt exists. Determine whether r nt a functin is cntinuus at a given pint. Recall and apply the cnditins under which mixed partial derivatives are equal. 16.1 Differentiability and Gradient differentiable multivariable functin gradient f a multivariable functin Evaluate the gradient f a functin. Find a functin with a given gradient. 16.2 Gradients and Directinal Derivatives directinal derivative isthermals Recall and prve identities invlving gradients. Give a graphical interpretatin f the gradient. Evaluate the directinal derivative f a functin. Give a graphical interpretatin f directinal derivative. Recall, prve, and apply the therem that states that a differential functin f increases mst rapidly in the directin f the gradient (the rate f change equal t the length f the gradient) and it decreases mst rapidly in the ppsite directin (the rate f change equal t the ppsite f the length f the gradient). Find the path f a heat seeking r a heat repelling particle. Slve applicatin prblems invlving gradient and directinal derivatives. 16.3 The Mean-Value Therem; The Chain Rule the Mean Value Therem fr functins f several variables nrmal line chain rules fr functins f several variables implicit differentiatin Recall and apply the Mean Value Therem fr functins f several variables and its crllaries. Apply an apprpriate chain rule t evaluate a rate f change. Apply implicit differentiatin t evaluate rates f change. Slve applicatin prblems invlving chain rules and implicit differentiatin. 16.4 The Gradient as a Nrmal; Tangent Lines and Tangent Planes
nrmal vectr tangent vectr tangent line tangent plane nrmal line Use gradients t find the fllwing fr a smth planar curve at a given pint. nrmal vectr nrmal line tangent line r tangent plane Use gradients t find the fllwing fr a smth surface at a given pint. nrmal vectr nrmal line tangent plane Slve applicatin prblems invlving nrmals and tangents t curves and surfaces. 16.5 Lcal Extreme Values lcal minimum and lcal maximum critical pints statinary pints saddle pints discriminant Secnd Derivative Test Find the critical pints f a functin f tw variables. Apply the Secnd-Partials Test t determine whether each critical pint is a lcal minimum, a lcal maximum, r a saddle pint. Slve wrd prblems invlving lcal extreme values. 16.6 Abslute Extreme Values abslute minimum and abslute maximum bunded subset f a plane r three-space the Extreme Value Therem Determine abslute extreme values f a functin defined n a clsed and bunded set. Apply the Extreme Value Therem t justify the methd fr finding extreme values f functins defined n certain sets. Slve wrd prblems invlving abslute extreme values.
16.7 Maxima and Minima with Side Cnditins side cnditins r cnstraints methd f Lagrange Lagrange multipliers crss-prduct equatin f the Lagrange cnditin Graphically interpret the methd f Lagrange. Determine the extreme values f a functin subject t a side cnditins by applying the methd f Lagrange. Apply the crss-prduct equatin f the Lagrange cnditin t slve extreme value prblems subject t side cnditins. Apply the methd f Lagrange t slve wrd prblems. 16.8-9 Differentials; Recnstructing a Functin frm its Gradient differential general slutin particular slutin cnnected pen set pen regin simple clsed curve simply cnnected pen regin partial derivative gradient test Determine the differential fr a given functin f several variables. Determine whether r nt a vectr functin is a gradient. Given a vectr functin that is a gradient, find the functins with that gradient. 17.1-3 Multiple-Sigma Ntatin; The Duble Integral ver a Rectangle; The Evaluatin f Duble Integrals by Repeated Integrals duble sigma ntatin triple sigma ntatin upper sum lwer sum duble integral integral frmula fr the vlume f a slid bunded between a regin Ω in the xy-plane and the graph f a nn-negative functin z=f(x,y) defined n Ω. integral frmula fr the area f regin in a plane integral frmula fr the average f a functin defined n a regin Ω. prjectin f a regin nt a crdinate axis
Type I and Type II regins reductin frmulas fr duble integrals the gemetric interpretatin f the reductin frmulas fr duble integrals Evaluate duble and triple sums given their sigma ntatin. Recall and apply summatin identities. Apprximate the integral f a functin by a lwer sum and an upper sum. Evaluate the integral f a functin using the definitin. Evaluate duble integrals ver a rectangle using the reductin frmulas. Sketch planar regins and determine if they are Type I, Type II, r bth. Evaluate duble integrals ver Type I and Type II regins. Change the rder f integratin f an integral. Apply duble integrals t calculate vlumes, areas, and averages. 17.4 The Duble Integral as the Limit f Riemann Sums; Plar Crdinates diameter f a set Riemann sum duble integral as a limit f Riemann sums plar crdinates [r,θ] transfrmatin frmulas between Cartesian and plar crdinates duble integral cnversin frmula between Cartesian and plar crdinates Represent a regin in bth Cartesian and plar crdinates. Evaluate duble integrals in terms f plar crdinates. Evaluate areas and vlumes using plar crdinates. Cnvert a duble integral in Cartesian crdinates t a duble integral in plar crdinates and then evaluate. 17.5 Further Applicatins f the Duble Integral integral frmula fr the mass f a plate integral frmulas fr the center f mass f a plate integral frmulas fr the centrid f a plate integral frmulas fr the mment f an inertia f a plate radius f gyratin the Parallel Axis Therem Evaluate the mass and center r mass f a plate Evaluate the centrid f a plate. Evaluate the mments f inertia f a plate. Calculate the radius f gyratin f a plate. Recall and apply the parallel axis therem.
17.6-7 Triple Integrals; Reductin t Repeated Integrals triple integral integral frmula fr the vlume f a slid integral frmula fr the mass f a slid integral frmulas fr the center f mass f a slid Evaluate physical quantities using triple integrals such as vlume, mass, center f mass, and mments f intertia. Recall and apply the prperties f triple integrals, including: linearity, rder, additivity, and the mean-value cnditin. Sketch the dmain f integratin f an iterated integral. Change the rder f integratin f a triple integral. 17.8 Cylindrical Crdinates cylindrical crdinates f a pint crdinate transfrmatins between Cartesian and cylindrical crdinates cylindrical element f vlume Cnvert between Cartesian and cylindrical crdinates. Describe regins in cylindrical crdinates. Evaluate triple integrals using cylindrical crdinates. 17.9 Spherical Crdinates spherical crdinates f a pint crdinate transfrmatins between Cartesian and spherical crdinates spherical element f vlume Cnvert pints and equatins between Cartesian and spherical crdinates. Describe regins in spherical crdinates. Cnvert integrals in Cartesian crdinates t integrals in spherical crdinates. Set up and evaluate triple integrals using spherical crdinates. 17.10 Jacbians; Changing Variables in Multiple Integratin Jacbian change f variable frmula fr duble integratin change f variable frmula fr triple integratin Find the Jacbian f a crdinate transfrmatin. Use a crdinate transfrmatin t evaluate duble and triple integrals.
18.1,4 Line Integrals; Anther Ntatin fr Line Integrals; Integrals with Respect t Arc Length wrk alng a curved path piecewise smth curve Evaluate the wrk dne by a varying frce ver a curved path. Evaluate line integrals in general including line integrals with respect t arc length. Evaluate the physical characteristics f a wire such as centrid, mass, and center f mass using line integrals. Determine whether r nt a vectr field is a gradient. Determine whether r nt a differential frm is exact. 18.2 The Fundamental Therem fr Line Integrals; Wrk-Energy Frmula; Cnservatin f Mechanical Energy path-independent line integrals gradient field Determine whether r nt a frce field is gradient field n a given regin, and if s, find its ptential functin. Recall, apply, and verify the Fundamental Therem fr Line Integrals. Identify when the Fundamental Therem f Line Integrals des nt apply. 18.5 Green's Therem Jrdan curve Jrdan regin Green's Therem Recall and verify Green's Therem. Apply Green's Therem t evaluate line integrals. Apply Green's Therem t find the area f a regin. Derive identities invlving Green's Therem 18.6 Parameterized Surfaces; Surface Area parameterized surface fundamental vectr prduct element f surface area fr a parameterized surface surface integral integral frmula fr the surface area f a parameterized surface integral frmula fr the surface area f a surface z = f(x, y)
upward unit nrmal Parameterize a surface. Evaluate the fundamental vectr prduct fr a parameterized surface. Calculate the surface area f a parameterized surface. Calculate the surface area f a surface z = f(x, y). 18.7 Surface Integrals surface integral integral frmulas fr the surface area and centrid f a parameterized surface integral frmulas fr the mass and center f mass f a parameterized surface integral frmulas fr the mments f inertia f a parameterized surface integral frmula fr flux thrugh a surface Calculate the surface area and centrid f a parameterized surface. Calculate the mass and center f mass f a parameterized surface. Calculate the mments f inertia f a parameterized surface. Evaluate the flux f a vectr field thrugh a surface. Slve applicatin prblems invlving surface integrals. 18.8 The Vectr Differential Operatr the vectr differential peratr divergence curl Laplacian Evaluate the divergence f a vectr field. Evaluate the curl f a vectr field Evaluate the Laplacian f a functin. Recall, derive and apply frmulas invlving divergence, gradient and Laplacian. Interpret that divergence and curl f a vectr fields physically. 18.9 The Divergence Therem utward unit nrmal the divergence therem sink and surce slenidal Recall and verify the Divergence Therem. Apply the Divergence Therem t evaluate the flux thrugh a surface. Slve applicatin prblems using the Divergence Therem.
18.10 Stkes' Therem riented surface utward, upward, and dwnward unit nrmal the psitive sense arund the bundary f a surface circulatin cmpnent f curl in the nrmal directin irrtatinal Stkes' therem Recall and verify Stke's therem. Use Stkes' Therem t calculate the flux f a curl vectr field thrugh a surface by a line integral. Apply Stke's therem t calculate the wrk (r circulatin) f a vectr field arund a simple clsed curve.