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# MATH-M312 Exam 1: All Notes and Exam Info Math M312

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This 24 page Study Guide was uploaded by Kathryn Brinser on Thursday February 11, 2016. The Study Guide belongs to Math M312 at Indiana University taught by Siddharth Bhaskar in Fall 2016. Since its upload, it has received 153 views. For similar materials see Calculus 4 in Mathematics (M) at Indiana University.

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M312 Section 5.4 and 5.5 Notes- Mean Values and Triple Integrals 1-15-16 2 Recall area of rectangle in ℝ : ???? ???? o For rectangle R with corners ????,???? and ????,???? , ???? =∬ ???? 1???????? = ∫????∫???? 1???????? ???????? 3 Analogous to volume of rectangular prism R in ℝ : o Corners (????,????,????), (????,????,????), and (????,????,????) o ???? = 1???????? = ???? ???? ???? 1???????? ???????? ???????? ∭ ???? ∫????∫????∫ ???? 1 ???? If ????:ℝ → ℝ, mean value of ???? over a line segment from ???? = ???? to ???? = ????????????????????ℎ ???????? ????????????????????????????????(????)???????? = 1 ???? ∫ ????(????)???????? ????−???? ???? If ????:ℝ → ℝ, mean value of ???? over a region R is1 ∬ ????(????,????)???????? ???????????????? ???????? ???????? o Suppose we have 1x3 rectangle: 1 and 2 are values of ???? Mean value not 1.5 because values not evenly spaced; average weighted by area Mean: 2(1 + 1(2 = 4 3 3 3 o In general, for any function ????:ℝ → ℝ (analogous to ℝ ) and domain ???? in ℝ (ℝ ), we can approximate mean value of ???? on ????: Partition space into finitely many rectangles For all inside ???? completely, approximate value of ???? in that box by value at center Edges are “pixelated” Mean value ≈ sum values of ???? at center of each box weighted by fractional area of each box ???????????????? ???????? ???? ≈ ∑ ???????????? ???????????????????? ???????????????????????? ???????? ???? ( ) ???? ???????????????????????????????????? ???????????????????? ???????????????? ???????? ???????????? ???????????????????? ???????? ???? 1 ∑ = ???????????????? ???????? ???????????? ???????????????????????????????? ????(???????????????????????? ???????? ????)(???????????????? ???????? ????) area of all boxes independent of ????, can be pulled out as constant Approaches mean as partitions get finer 1 lim ( ∑???????????????????? ???? ???????????????????????? ???????? ???? ???????????????? ???????? ???? )) # ????????????????????→∞???????????????? ???????? ???????????? ???????????????????? 1 = ???????????????? ???????? ???????? ????(????,????)???????? 1 If ????:ℝ → ℝ, mean value is ∭ ????(????,????,????)???????? ????????????. ???????? ???????? M312 Chapter 5 Notes- Review of Double/Triple Integrals 1-13-16 Integration in “Euclidean space”- in ℝ, ℝ , and ℝ In 3 variables, if D is interior of some region of space, then its boundary is its surface (ie. ellipsoid) o Interior 3D, boundary 2D o Boundary of 2D surface is 1D (curve) o And so on; boundary of object in ???? dimensions has ???? − 1 dimensions Generalization of Fundamental Theorem o ∭ ???? = ∬ ????????ℎ???????? ???????????????????????????????? (Stokes’ Theorem*) o ???? = ????????ℎ???????? ???????????????????????????????? (Green’s Theorem*) *clarify ∬ ???? ∫ o ∫???? ???? = ???? ???? − ????(????) where antiderivative is “0-dimensional” integral Riemann Sums for 2-variabl2 functions o Suppose D is a region of ℝ o Partition D horizontally and vertically; do not have to be evenly spaced o There are infinite boxes around boundary; “pixelated” 2 o Suppose ????:ℝ → ℝ; Riemann Sum of ???? over D with respect to the partitions is sum ???? evaluated at center (or some point) in each box completely contained in D multiplied by area of that box ∬???? ????(????,????)???????? = # ????????????????????→∞∑???????????????????? ???????????????????????????????????????????????????????? ???????? ???????????? ∙ (???????????????? ???????? ???????????? ) For all functions we will see, Riemann Sums converge to a number when making finer and finer partitions of D; limit is definition of double integral Triple Integrals o Suppose D is a region of space in ℝ and ????:ℝ → ℝ o Divide space into thin 3D boxes/prisms in 2 perpendicular directions (typically parallel to ???????? and ???????? planes) o Make boxes infinitely thin (into planes) and increase number, let it approach ∞ o o ∭ ????(????,????,????)???????? = lim ∑???????????????????????? ???????????????????????????????????????????????????????? ???????? ????????????????????????∙ (???????????????????????? ???????? ????????????????????????) ???? # ????????????????????????→∞ Kinds of Regions Type I Not Type I Type II Not Type II ????-simple regions in ℝ - new name for Type I regions o Area between 2 functions of ????; set of all points ????,???? such that ???? < ???? < ???? and ???? ???? < ???? < ℎ ???? o If D is y-simple region,∬the????(????,????)???????? = ∫????=???? ∫????=ℎ(????)????(????,????)???????????????? ???? ????=???? ????=????(????) ????-simple regions in ℝ - new name for Type II regions o Area between 2 functions of ????; set of all points (????,????) such that ???? < ???? < ???? and ???? ???? < ???? < ℎ(????) o If D is x-simple region,∬the????(????,????)???????? = ∫????=???? ∫????=ℎ(????)????(????,????)???????????????? ???? ????=???? ????=????(????) Simple regions- both Type I and Type II o 6 ways to order variables for ????(????,????,????) in simple region in ℝ Consider one representation: set of all points ????,????,???? such that ???? < ???? < ????, ???? ???? < ???? < ℎ ???? , and ????1????,???? < ???? < ???? ????2???? ) NOTE: plane boundaries should follow ????(????) and ℎ(????) Get “tube” shape when you extend region bounded by ???? and ???? on ???????? plane in ±???? directions Draw in ????1and ????2above ???????? plane to get a 3D bounded region For this ordering of variables where D is a simple region defined by1????,2,????,ℎ,???? ,???? , then ????=???? ????=ℎ(????) ????=????1(????,????) ∭ ???? ????(????,????,????)???????? = ∫????=???? ????=????(????) ????=????1(????,????)(????,????,????)???????????????????????? Ex. Express the unit ball (unit sphere including area inside) as a simple region. o −1 < ???? < 1 −√1 − ???? < ???? < √1 − ???? 2 −√1 − ???? − ???? < ???? < √ 1 − ???? − ???? 2 o ???? bounded between constants, ???? between upper and lower halves of unit circle; this alone would make a cylinder, because nothing depends on ???? at this point o ???? bounded between upper and lower hemispheres of unit sphere Ex. Center a cylinder of radius 1 around the ????-axis. Do the same thing around the ????-axis. Express the interior of the intersection as a simple region. o Both cylinders infinite; intersection finite Intersection: o Cross sections with ???????? and ???????? planes are unit circles 2 2 o −1 < ???? < 1 −√1 − ???? < ???? < √1 − ???? o Find “capping” functions that bound ????: Upper and lower halves of cylinder centered around ????-axis (in ???????? plane) ????1????,???? = lower half, 2 ????,???? = upper half Equation of cylinder: ???? + ???? = 1 because value does not depend on ???? at all; can be anything 2 2 ????1????,???? = −√1 − ???? ????2????,???? = √1 − ???? So −√1 − ???? < ???? < √1 − ???? 2 If D is region in ℝ ,∬the1???????? = area of D; if D is region in ℝ ,∭the1???????? = volume of D ???? ???? Ex. Find the volume of the interior of the intersection of the two unit cylinders from the previous example. The region will be called D. o ???? = ∭ 1???????? ???? 2 2 = ∫ ∫ √1−???? ∫ √1−???? 1???????????????????????? −1 −√1−????2 −√1−????2 1 √1−????2 √1−????2 = ∫−1 −√1−????2( ????|−√1−????2)???????? ???????? 1 √1−????2 = ∫−1 −√1−????22√1 − ???? ???????????????? √1−????2 = ∫1 2????√1 − ???? 2| ???????? −1 −√1−????2 = ∫1 2( (1 − ???? √) − ???? − − √1 ( ???? √1 −)???? )???????? 2 −1 = 1 4 1 − ????2)???????? ∫−1 = 4∫ 1 1 − ???? ???????? −1 1 3 1 = 4(???? − 3 )| −1 1 1 = 4(1 − 3 − 4(−1 + ) 3 2 2 = 4( ) − 4(− ) 8 38 16 3 = + = 3 3 3 M312 Section 6.1 Notes- The Geometry of Maps from ℝ to ℝ 2 1-20-16 2 Map/transformation- suppose ????:ℝ → ℝ; for each point, outputs another point Ex. ???? ????,???? = ????cos????,????sin???? ) ???? o (1,???? → 1cos????,1sin???? = −1,0 ( ) o If ???? > 0, distance from origin is ???? o Looks like polar coordinates; ????0???? 0???? ) is point with distance ???? from origin and angle ???? wi????-axis o Ex. If ???? ,???? )on ????-axis, each point is (????,0). We say that ???? fixes the ????-axis, because ???? ????,0 = 0 0 ???? (????cos0,????sin0 = ????,0 . What does ???? do to the line ???? = ? 4 ???? Points on line are (????4 ) ???? ???? ???? ????(????, )4= (????cos ,????s4n ) 4 Applying transformation, ???? maps points on ???? = ????to line ???? = ???? because sin = cos while ???? 4 4 4 varies; each point is2(????,???? with distance ???? from origin 2 Ex. Take the rectangle 0,0 × 1,2???? and apply ???? ????,???? = ????cos????,????sin???? to all points on it. What shape results? o Rectangle ???? = set of all points (????,????) such that 0 ≤ ???? ≤ 1 and 0 ≤ ???? ≤ 2???? o In case of negative radii: ????,???? → −????,???? + ???? where first ???? < 0 o Hence as (????,????) ranges over ????, ???? ????,???? = ????cos????,????sin???? ranges over all points whose distance from origin is between 0 and 1 and whose angle with origin is between 0 and 2π o ???? maps unit circle from ????- notated as ???? ???? = unit circle; apply transformation ???? to all points in region ????, region (unit circle) is obtained ???? is the pre-image of the unit circle (image) Notice that all points with ???? = 0 map to origin; not just one point maps to it Also notice that all points with ???? = 0 and ???? = 2???? map to line on ????-axis from radius 0 to 1 o Ex. For which domain ???? is ???? ???? a circle with radius ????? If we extend ???? to have horizontal length ????, radius of circle increases to ????; range of ???? values to transform increases from 0 ≤ ???? ≤ 1 to 0 ≤ ???? ≤ ???? o Ex. For which domain ???? is ???? ???? the ring between the unit disk and a disk with radius ? 2 Move left boundary from 0 to ½, so ???? maps a circle with the smallest radius being ½ instead of 0; recall how original disk had all points with radius 0 mapped to origin o Ex. What is the “smallest” ???? such that ???? ???? is ℝ (the ???????? plane)? Infinite circle covers all of ℝ - need infinite box ???? Extend right side to infinity (and beyond) Other changes to ???? that affect how ???? maps: o Moving top edge down: Cuts out slice; 0 ≤ ???? ≤ ???? < 2???? o Moving top edge up: Re-traces parts of circle twice; maximum angle becomes greater than 2???? Image of circle unchanged o Crossing axes: Easier to see with sector of circle; radius becomes negative, sector reflects over ???? and ???? axes One-to-One (1-1) Transformations o No 2 points in domain/pre-image map to same point in range/image o If ????0,????0) ≠ ???? ,1 ,1then ???? ???? ,????0 0) ≠ ???? ???? 1???? 1) Alternatively, if ???? ???? ,???? = ???? ???? ,???? , then ???? ,???? ) = ???? ,???? ) 0 0 1 1 0 0 1 1 2 o Functions can be 1-1 on specified domain; for pre-image region ???? ???? ℝ , we say ???? is 1-1 if no 2 points in ???? map to same point in image o Ex. ???? ????,???? = ????cos????,????sin???? not 1-1 because ???? ???? ,???? ( )= ???? ???? ,???? + 2???? ) 0 0 0 0 There are points in ???? on ????-axis, which all map to origin of unit disk Removing boundary of ???? where ???? = 0 results in rectangle 0,1 × 0,2???? ] Still not 1-1, because every point on top border (???? = 2????) maps to same line in unit disk (from origin to 2????,0 ) as every point where ???? = 0 Final result: unit disk has lost origin and point 2????,0 ; still has line between those points because it is covered by points with angle 0 Show that no two points map to the same point when ???? is 0,1 × 0,2???? (the rectangle ????). In other words, show ???? is 1-1 on ????, where ???? 0???? 0 ???? ,1 1)???? ????, and suppose ???? ???? ,???? ) = ???? ???? ,???? ) 0 0 1 1 Since points are in ????, we know 0 < ????0≤ 1, 0 < ???? ≤11, 0 ≤ ???? < 20, 0 ≤ ???? < 2???? 1 By assumption, since ???? ???? 0???? 0) = ???? ???? 1???? 1) o (????0cos???? 0???? s0n???? 0) = ???? c1s???? ,????1si1???? 1) o ???? cos ???? + ???? sin ???? = ???? cos???? + ???? sin ???? 2 2 2 2 0 0 0 1 1 1 1 o ????0= ???? 1 o ????0= ???? 1 because both must be positive Now we know, since ???? = ???? : 0 1 o ????0cos???? 0 ???? c0s???? 1 o Since 0 > 0, can cancel it and get co0???? = cos????1 o Similarly, si0???? = sin????1 o Since ???? and ???? give identical cosines, they must differ by an integer multiple of 2???? 0 1 o Since 0 ≤ ????0,????1,< 2????, we know ???? 0 ???? 1 ∴ ???? 0???? 0) = ???? 1???? 1 and no two points map to the same point when ???? is 1-1 on the rectangle ???? with the left and top boundaries removed Linear Transformations o We say a transformation/map ????:ℝ → ℝ is linear when ???? ???? + ????1 2) = ???? ????1 )+ ???? ???? 2)and ???? ???????? = ???????? ???? for any ???? ???? ℝ and ???? ???? ℝ 2 3 o Most common in ℝ and ℝ o If linear map ????:ℝ → ℝ , then ???? completely determined by ???? 0,1 and ???? 1,0 ) Any vector (????,????) can be written as linear combination of 0,1 and 1,0 : (????,???? = ???? 1,0 + ???? 0,1 ) ???? ????,???? = ???? ????(1,0 + ???? 0,1( )) = ???? ( 1,0 )) + ????(???? 0,1 )) = ???????? 1,0 + ???????? 0,1 ) If we know ???? 1,0 and ???? 0,1 , we know ???? ????,???? for any ????,???? ???? ℝ 2 These vectors form a basis o General rule- linear map completely determined by where it sends a basis o Every linear transformation (LT) has matrix representation Show that for any LT ????, there is a matrix ???? such that ???? ???? = ???????? Suppose ???? 1,0 = (????,????), ???? 0,1 = ????,????( ) ???? ????,???? = ???? ????,???? + ???? ????,????( ) o = ???????? + ????????,???????? + ???????? ) ???? ???? ???? o = [???? ???? ???? ] What have we found out? 2 2 If ???? is linear map ℝ → ℝ , then ???? ???? = ???????? where ???? is matrix with first column ???? 1,0 and second column ???? 0,1 ) ???? is matrix representation of ???? ????+???? ????−???? o Ex. Let ???? ????,???? = ( 2 , 2 ). Show that ???? is linear (give its matrix representation). 1 ???? ????,???? = 2(???? + ????,???? − ???? ) 1 ???? + ???? = 2 ???? − ????) 1 1 1 ???? = [ ][????] 2 1 −1 ???? has a matrix representation, so it is linear Where does ???? send the square −1,1 × −1,1 with its interior? Label vertices: ???? = (1,1) ???? = −1,1 ) ???? = −1,−1 ) ???? = 1,−1 ) New vertices: ???? ???? = 1,0 ) ???? ???? = 0,−1 ) ???? ???? = −1,0 ) ???? ???? = 0,1 ) Edges preserved in transformation: o Edge between ???? and ???? is line segment ???? + ???? ???? − ???? for 0 ≤ ???? ≤ 1 o Now consider ????(???? + ???? ???? − ???? )) for 0 ≤ ???? ≤ 1: = ???? ???? + ???? ????(???? − ???? ???? ( ) ) Since ???? is linear 0 ≤ ???? ≤ 1, this is line segment between ????(????) and ????(????) o ∴ linear ???? maps line segments to line segments Final figure: o In general, map endpoints and put edges in vector line form like example above; maps take edges to edges between corresponding points o Linear maps take interiors of finite regions to interiors of finite regions o What do linear maps do to polygons? Look at, label, apply ???? to vertices Connect corresponding edges (ie. If ???? and ???? share an edge, connect ???? ???? → ???? ???? Interior of new figure is image of original filled-in polygon Intuitively, # vertices in pre-image = # vertices in image 2 Any LT in ℝ can be expressed as combination of rotations, reflections, and dilations o Not analogous to ℝ , etc. In ℝ , transformations parameterized by a plane (reflect around plane), rotate around line parallel to normal vector of plane o Dilation- scalar multiplication; ???? ???? is dilation if ???? ???? = ???????? for some ???? ???? ℝ where ???? = 2 To find matrix, start with basis vectors (1,0) and (0,1) represented in iden[ity m],rix 0 1 multiply by ???? Ex. Stretch the triangle with vertices (3,2), (−3,2), and 3,−2 by a factor of 2 in the ???? direction. Multiply identity matrix by ???? = 2 for ????-coordinate only: ???? =2 o [1 0] →???? [1 0 ] 0 1 0 2 Recall that ???? ????,???? = ???????? for some matrix ????: o ???? 3,2 = [1 0 ][ ]= ( 3 + 0 2 ,0 3 + 2 2 ( ) )= 3,4 ) 0 2 2 o ???? −3,2 = [1 0 ][ ] = ( −3 + 0 2 ,0 −3 + 2 2 ( )) = −3,4 ) 0 2 2 o ???? 3,−2 = [1 0 ][ 3 ] = ( 3 + 0 −2 ,0 3 + 2 −2( )) = 3,−4 ) 0 2 −2 ( ) 2 2 o Rotations by ????- ???? ???? :ℝ → ℝ is ???? rotated counterclockwise through angle of ???? Why are rotations linear? (proof by picture) Show for vectors ???? and ???? and real number ???? that ???? ???? ???? + ???? = ???? ???? + ???? ???? a????d ) (???????? ???????????????? = ???????? ???? ????( ) (i) By parallelogram rule, rotation of added vectors is same as addition of rotated vectors (ii) By observation, rotation of scalar multiple of vector is same as scalar multiple of rotation of vector To find matrix, start with basis vectors Specify new vectors by trigonometry: o o ???? 1,0 = cos????,sin???? ) ???? o ????????0,1 = −sin????,cos???? ) ???? ???? = [ cos???? −sin???? ][ ] ???? sin???? cos???? ???? Ex. Rotate the vector ???? counterclockwise by ???? = 45°. ???? ???? cos 4 −sin 4 ???? ???? 4 = [ ???? ???? ][????] sin4 cos4 √2 −√2 2 2 ???? = [√2 √2 ][????] 22 2 2 2 2 = (√ ???? − √ ????,√ ???? + √ ????) 2 2 2 2 √2 = 2(???? − ????,???? + ???? ) Ex. Using the rotation above 45° , what is the image of the square 1,2 × 0,1 under ????? Notice that square’s location is based on position vector 1,0 - coordinates of vector not significant here, but it will rotate with the square by 45° Vectors that specify each corner get rotated, so whole figure gets rotated o Reflections over ????- ???????? ???? is reflection of ???? having angle ???? with ????-axis; that becomes your “mirror” ???? Why are reflections linear? (proof by picture) Show for vectors ???? and ???? and real number ???? that ???? ???????? ???? + ???? = ???????? ???? +???????????? ???? and ????( ) (???????? ???????????????????? = ???????????? ???? ???? ( ) ???? (i) Consider ???? =2; reflecting sum of vectors results in same thing as sum of reflections of individual vectors (ii) By observation, reflection of scalar multiple of vector is same as scalar multiple of reflection of vector Often more difficult: to find matrix representation, can use identity matrix Reflections can also be represented with projections or rotations Projections: o o Middle vector ???? (parallel with line of angle ????) is a projection of ????; if you can find that vector (you can), you can fin????∙????ifference between projection????∙????d ???? Recall: ???????????????????????????????? = ???????? = ????, and ???? − ????????????????????????????(???? = ???? − ???? = difference ????∙???? ????∙???? vector we want o Add difference vector to ???? twice to get reflection ????∙???? ????∙???? ???????? ???? = ???? + (???? − ????∙????????) = 2???? − ????∙???? o Ex. Find ???????? (????) when ???? = 5,1 and ???? = . ???? ???? 4 ???? ( ) ???? ???? √2 √2 ( ) Find an equation for ???? =4: 0,0 + ????(cos ,4in ) 4 ????( 2 ,2 ) → ???? 5,5 ???? = ???????????????? (5,1 = (5,1 ∙ 5(5,5) (5,5 (5,5 ∙ 5,5 = 25+5 (5,5) 25+25 = 30 5,5 ) 50 = 3,3 ) ???? − ???? = 3,3 − 5,1 = −2,2 ( ) Shift −2,2 from origin to end of ????; add twice ???? + 2 −2,2 = 5,1 + −4,4 = 1,5) ( ) Intuitively, reflection over ???? = is inverse of original, as we found 4 Rotations: o Rotate ???? so ???? = 0 cos −???? ) −sin −???? ) ???? ???? −????(???? = [ ][ ] = ???? ???? sin −???? ) cos −???? ) ???? o Find matrix for reflection across ????-axis (flip ????-coordinate) 1 0 ???? ???????? 0( ????) = [0 −1 ][????] = ???? ???? o Rotate ???? back so ???? is at its original position ???? ???? ????( ????) = [cos???? −sin???? ][ ]= ???? ???? sin???? cos???? ???? ???? o Ex. Reflect the point 3,2 over the line ???? = . 3???? 3???? 2 cos −sin ???? −????(3,2 = [ 3???? 3????2][ ] sin cos 2 2 2 = [ 0 1 ][ ] −1 0 2 = 2,−3 ) 1 0 2 ???????? 0,−3 = [ 0 −1 −3][ ] = 2,3 ) ???? ???? cos −sin 2 ???? ????,3 = [ ????2 ????2][ ] sin2 cos2 3 0 −1 2 = [ ][ ] 1 0 3 = (−3,2) Derivatives of Transformations in ℝ o ???? can be represented as ???? ,???? where ???? ,???? : ℝ → ℝ 1 2 1 2 If ???? ????,???? = ????cos????,????sin???? , then ???? ????,1 = ????cos???? and ???? ????,???? = 2sin????) o Derivative of ???? (???????? or ???? ) is matrix (just accept it): ????????1 ????????1 ???????? ???????? −∇???? −1 [????????2 ????????2]= −∇???? − ] ???????? ???????? 2 cos???? −????sin???? o Ex. When ???? ????,???? = ????cos????,????sin???? , ???? = ) ′ [ ] sin???? ????cos???? Notice that if we take determinant of ????′, we have these observations: ????:ℝ → ℝ , det????′:ℝ → ℝ2 2 2 det????′ = ???????????????? ???? − −???????????????? ???? = ???? ) o Ex. Find the determinant of ????′ when ????:ℝ → ℝ and is given by ???? ????,????,???? = ????????,????????,???????? . ) ????????1 ????????1 ????????1 ???????? ???????? ???????? ???????? ???????? ???????? ???? = 2 2 2 ???????? ???????? ???????? ????????3 ????????3 ????????3 [???????? ???????? ???????? ] 0 ???? ???? = [ ???? 0 ????] ???? ???? 0 0 ???? ???? ???? ???? 0 det????′ = 0 |???? 0 |− ???? ???? 0| + ???? ???? ????| = 0 − ???? 0 − ???????? + ???? ???????? − 0 ) = ???????????? + ???????????? = 2???????????? M312 Section 6.2 Notes- The Change of Variables Theorem 1-27-16 Change of Variables in ℝ2 o In transformations, typically go from ???????? plane to ???????? plane where ???? = ???? ????,???? and ???? = ???? ????,???? o Advantageous to use ????????????????, etc. in integrals instead of ???????? to show which plane you are working in o Change of variables generalizes concept of ????-substitution o Reasons to change variables: Integrand itself too difficult to evaluate Region/domain hard to integrate over o Suppose ????:ℝ ???????? → ℝ ???????? ; let ????,???? = ???? ????,???? so ???? and ???? are functions of ???? and ???? o Ex. ???? ????,???? = (???? ????1???? ,???? ????,2 ) = (???? ????,???? ,???? ????,???? ) = ???? + ????,???? − ???? ) ????1 ???????? Instead of???????? or similar notations, can u????????, etc. o Say we want to integrate∬???? ????(????,????)???????????????? in ???????? plane: Let ???? be pre-image of image ???? under ???? such that ???? ????∗] = ???? ???????? ???????? ∗ ( ) ( ) ???????? ???????? With assumption that ???? is 1-1 on ???? ∬ ???? ????(????,????)???????????????? = ∬ ????∗????(???? ????,???? ,???? ????,???? )| ???????? ????????|???????????????? where ???????? ???????? ???????? ???????? |???? ????????| = det ????′ = Jacobian Determinant (used as correction factor; chain rule too?; SHOULD ALWAYS ???????? ???????? ???????? ???????? BE POSITIVE- bars also mean absolute value in integrals) o “Coordinate-free” formula for change of variables: ′ ∬???? ????∗]???? ???????? = ∬ ????∗(???? ∘ ???? det???? ???????? o Derivative of linear map is constant: Ex. If ???? = 3????, ???? = 3 1 2 ???? 1 2 If ???? = [ ][ ], ???? = [ ] 3 4 ???? 3 4 Proof: ( ) ???? = ???? + 2????,3???? + 4???? ????????1= 1 ????????1 = 2 ????2 = 3 ????2 = 4 ???????? ???????? ???????? ???????? ???????? ???????? ′ ???????? ???????? 1 2 ???? = [ ???????? ???????? ] = [ ] 3 4 ???????? ???????? o Ex. Let ???? be the parallelogram (including the interior) with vertices 0,0 , 1,2 , 2,2 , 3,4 . Evaluate ∬???? ???????? ????????????????. Both type I and type II region Doable as regular double integral, but change of variables makes domain of integration simpler We know ???? maps ????-gons to ????-gons; easy pre-image would be rectangle 0,1 × 0,1 ] Is there a transformation ???? that takes/maps 0,1 × 0,1 to ????? Assume ???? is linear, that the following are true, and solve for ????: o (0,0 → 0,0 ) linear transformations always map origin to origin o (1,0 → 2,2 ) o (0,1 → 1,2 ) o (1,1 → 3,4 ) 2 1 ???? From trial and error, ???? ????,???? = [ ][????] = 2???? + ????,2???? + 2???? correctly maps vertices of square to 2 2 those of ???? On 0,1 × 0,1 = ????, ???? is 1-1 ???? is 1-1 if and only if det????′ ≠ 0 Here, 2 2 − 2 1 = 2 ≠ 0 By change of variables∬ ???????? ???????????????? = ∬ ???? ????,???? ∙ ???? ????,???? det???? ′)???????????????? ???? ???? ( )( ) 2 1 = ∬ ???? 2???? + ???? 2???? + 2???? | 2 2|???????????????? recall that ????′ is constant matrix if ???? linear 2 2 = ∬ ???? (4???? + 4???????? + 2???????? + 2???? )(4 − 2 ???????????????? = 2 ∬ 4???? + 6???????? + 2???? ???????????????? 1???? 1 = 2 ∫ ∫ 4???? + 6???????? + 2???? ???????? ???????? 0 0 1 = 2 ∫1 ( ???? + 3???? ???? + 2???????? | )???????? 0 3 0 1 4 2 = 2 ∫0 3+ 3???? + 2???? ???????? 4 3 2 1 = 2( ???? + ???? + ???? | ) 3 3 2 3 0 4 3 2 = 2( 3 + 2 3 8 9 4 = 3 + 3 3 21 = 3 = 7 Change of Variables to Polar Coordinates ???????? ???????? ???????? ???????? o Reformat original formula∬ ???? ???? ????,???? ???????????????? = ∬ ????∗????(???? ????,???? ,???? ????,???? )| ???????? ????????|???????????????? where ????,???? =) ∗ ???????? ???????? ???? ????,???? and ???? ????[ ]= ???? o If ???? ????,???? = ???? cos???? ,???? sin???? (standard rectangular-to-polar transformation), notice that Jacobian reduces to ????: This is why polar integrals always have ???? ???????????????? |cos???? −???? sin???? | = ???? cos ???? − −???? sin ???? = ???? sin???? ???? cos???? o Ex. Compute the area of an ellipse ???? with semi-axes ???? and ???? using a change of variables. ???? = ∬???? 1???????????????? Is there a transformation ???? such that the pre-image of ???? under ???? is a box? ???? ????,???? = (???? ????,???? ,???? ????,???? ) = ???? cos???? ,???? sin???? maps the box ???? = 0,1 × [0,2????] to the unit circle, though polar, not linear Goal: change that box into ???? when mapped To make circle of radius ????, multiply ???? ????,???? and ???? ????,???? by ????: ???? ????,???? = ???????? cos???? ,???????? sin???? To get different semi-axes, multiply ???? ????,???? by ???? and ???? ????,???? by ????: ???? ????,???? = ???????? cos???? ,???????? sin???? ) ∴ ???? ????,???? = ???????? cos???? ,???????? sin???? maps ???? to ???? ???? not 1-1, but change of variables formula still holds if boundaries cause ???? not to be 1-1 ???? correctly maps ???? to ????: Think of ???? as composition of 3 transformations starting with original that results in unit disk Can factor ???? = ????3∘ ????2∘ ????1where ???? 3,???? = ???? cos???? ,???? sin???? ,???? ????,2 = ????????,???? ,( ) ????3????,???? = (????,????????) ???? ????,???? = ???? (3 (2 ????1???? )) ) ????1maps ???? to unit disk; 2 stretches ????-axis by ???? by scalar multiplica3ion; ???? stretches ????-axis by ???? ???? not 1-1 on ????- maps everything on ????-axis to origin,everything with ???? = 0 and 2???? to semi-axis ???? Ignoring boundaries, ???? is 1-1 on interior of ???? ???? = ∬ 1???????????????? = ∬ 1|???? cos???? −???????? sin???? |???????????????? ???? ???? ???? sin???? ???????? cos???? 2 2 = ∬ ???? ???????????? cos ???? − −???????????? sin ???? ???????????????? = ∬ ???? ???????????? ???????????????? 1 2???? = ∫0∫ 0 ???????????? ???????? ???????? use limits of ????; remember that it is same even when working with ellipse 1 2???? = ???????? ∫0 ????????|0 ???????? 1 = 2???????????? ∫0 ???? ???????? 1 1 = 2???????????? ( ???? | ) 2 0 = ???????????? Without change of variables: ???? ???? ???? 1− ???? ???? = −???? ∫ ????21???????? ???????? not fun. a drag. −???? 1−????2 o Ex. Let ???? be a quarter-ring with radii ???? and ???? (inner and outer). Use a change of variables to polar coordinates 2 2 to compute ∬ ???? ln ???? + ???? ) ????????????????. NOTE: ???????????? now means ???????? unless specified otherwise ???? To get quarter-ring, use rectangle 0,1 × 0,2???? - restrict ???? ≤ ???? ≤ ???? and 0 ≤ ???? ≤ 2 ???? ???? = ???? where ???? is ????,???? × [0, ] ???? 2 Notice ???? is 1-1 on ????, and ???? and ???? are functions of ???? and ????: ???? ????,???? = ???? cos???? ,???? sin???? ∬ ln ???? + ???? 2) ???????????????? = ∬ ln ???? + ???? 2)???? ???????????????? ???? ???? ( 2) = ∬ ???? ln ???? ???? ???????????????? ???? ???? 2 = ∫????∫ 02ln ???? )???? ???????? ???????? ???? = ???????????? ln ????2)| ???????? ∫???? 0 ???? ???? ( 2) = 2∫???? ???? ln ???? ???????? ???? ???? = 2∫???? 2???? ln(????)???????? integration by parts; let ???? = ln???? ,???????? = ???? ???????? ???? = ???? ( ???? ln????| − 1∫ ???????? ( )????????) 2 ???? 2 ???? ???? 1 ???? 1 ???? = ???? ( ???? ln????| − ∫ ???? ????????) 2 ???? 2 ???? ???? = ???? ( ???? ln???? − ???? )| 2 4 ???? 1 1 1 = ???? ( ???? ln???? − ???? ln???? − (???? − ???? )) 2 2 4 ???? 2 2 1 2 1 2 = (???? ln???? − ???? ln???? − ???? + ???? ) ???? 1 2 2 1 = [(???? ln???? − ???? ) − (???? ln???? − ???? )] 2 2 2 2 The Gaussian Integral ∞ 2 ???? 2 o ∫ ????−???? ???????? = lim ∫ ????−???? ???????? −∞ ????→∞ −???? −????2 o Used in statistics; ????is curve for normal distribution (area is finite) o Requires change of variables despite being single integral o How can we make sense of ∬ ???? ????????? ℝ2 Let ????????be disk with radius ????- can define integral as limit of ????rea of ???? as ???? → ∞ (increasingly large disk): ∬ ???? ???????? = lim ∬ ???? ???????? ℝ 2 ????→∞ ???????? Let ???? be rectangle −????,???? × −????,???? - can define integral as limit of area of ???? as ???? → ∞ (increasingly ???? ???? large box)∬ ℝ2???? ???????? = ????→∞ ∬ ???????????? ???????? Does not matter which shape; under reasonable assumptions, each limit gives same value Directly analogous to ℝ ,…,ℝ o Consider∬ ???? −(???? +???? ????????????????: ℝ2 −(???? +???? ) Evaluate using ???? : li∬ ???????????? ???????????????? ????→∞ 2 2 First chang∬ ????−(???? +???? )???????????????? to polar coordinates: pre-image of ???? is 0,???? × 0,2???? ???????? ???? ???? 2???? −????2 = ∫0∫ 0 ???? ???? ???????? ???????? ???? 2 2???? = ∫ ???? −???? ????????| ???????? 0 ???? 20 = 2???? ∫ ????−???? ???? ???????? 0 ???? −1 −????2 = 2???? ( 2 ???? )| 00 1 −????2 = 2???? (2???? )| 2???? = ????(1 − ???? −???? ) 2 lim ????(1 − ????−???? ) ????→∞ 1 = ????→∞ ???? (1 − ????????2) = ???? 1 − 0) = ???? 2 2 Evaluate using ???? : li∬ ????−(???? +???? )???????????????? ???? ????→∞ ???????? −(???? +???? ) ???? ???? −(???? +???? ) ∬???? ???? ???????????????? = −???? −???? ???? ???????? ???????? ???? ???? ???? 2 2 o = ∫ ∫ ????−???? ????−???? ???????? ???????? ???????? −???? 2 ???? 2 o = ∫ (????−???? (∫ ????−???? ????????)????????) −???? 2 −???? 2 2 o = ∫???? ???? −???? ???????? ∫???? ???? −???? ???????? since ???????? is constant in ????, can pull it out of integral; same −???? −???? with ????−????2inside new integral with respect to ????;result is product of two integrals ???? −????2 2 o = (∫−???????? ????????) renamed variable- both integrals are the same Take limit as ???? → ∞ of both sides: −(???? +???? ) ???? −????2 2 o lim ∬ ???????????? ???????????????? = lim( ∫−???? ???? ????????) ????→∞ ????→∞ 2 o ∬ ????−(???? +???? )???????????????? = ( ∫∞ ???? −????????????) ℝ 2 −∞ o We have shown with ???? that left side = ???? 2 ???? o ???? = (∫∞ ????−????2????????) −∞ o ∞ ????−????2???????? = ???? must be positive; cannot have negative area ∫−∞ √ Change of Variables in ℝ ∗ ∗ o If ????:ℝ → ℝ, ????:ℝ????→ ℝ ,???????? ???? ℝ ???? ???? ???? ℝ ???? ???? ????] = ????: ???????? o ∫???? ???? ???? ???????? = ∫????∗????(???? ???? )| ????????|???????? o Why absolute value here? Typical u-sub does not have it 1 2 ???????? Ex. ∫0 3???? ???????? let ???? = −????,???????? = −1 0 2 = − ∫1 3???? ???????? u-sub flips limits on integral when ???? = −???? instead of ????; unique to single integrals This is why ???????? ???? ???????? = − ???? ???? ???? ???????? ∫???? ∫???? Change of Variables in ℝ 3 3 3 2 o If ????:ℝ → ℝ, ????:ℝ ???????????? → ℝ ????????????, analogous to ℝ o ∭ ???? ???????? = ∭ ∗(???? ∘ ???? det???? ???????? where ????,???? ???? ℝ , ???? ???? ∗]= ????, and ???? is 1-1 on ???? except maybe on ???? ???? boundaries ???????? ???????? ???????? ???????? ???????? ???????? ???????? ???????? ???????? o ∭ ???? ????,????,???? ???????????????????????? = ∭ ∗????(???? ????,????,???? ,???? ????,????,???? ,???? ????,????,???? ) ) | |???????????????????????? ???? ???? ???????? ???????? ????????| ???????? ???????? ???????? ???????? ???????? ???????? Change of Variables to Spherical Coordinates o 3-dimensional analog to polar; based on ????, ????, ???? ???? = angle on ???????? plane from ????-axis; specifies plane through ???? axis ???? = angle of “tilt” outward from ????-axis ???? = distance outward from origin (based on where ???? and ???? place point) o Ex. Graph the rectangular point (0,1,1) in spherical coordinates. ???? ???? (????,????,???? = ( 2√ , 4 2 ( ) o Ex. Given ????,????,???? , what are its rectangular coordinates? 2 different representations of point (at top of blue triangle): By geometry, can figure out coordinates: ∴ ????,????,???? = ???? sin???? cos???? ,???? sin???? sin???? ,???? cos???? ) Notice: all points in ℝ have spherical coordinates such that 0 ≤ ???? ≤ 2???? (any angle in ???????? plane) and 0 ≤ ???? ≤ ???? (when ???? = 0, points in positive ???? direction; when ???? = ????, points in negative ???? direction- only need 180 degrees’ worth for this angle, because there are other ways to achieve the same angle with different values of ????) o Ex. Consider the spherical coordinate transformation ????:ℝ ???????????? → ℝ ???????????? given by ???? ????,????,???? =) (???? sin???? cos???? ,???? sin???? sin???? ,???? cos???? . What is the pre-image of the unit ball under ????? ie. What points in ????,????,???? space get mapped into unit ball in ????,????,???? space? Restrict distance from origin: 0 ≤ ???? ≤ 1 ???? can be any angle disregarding multiples of 2????: 0 ≤ ???? ≤ 2???? ???? can be any angle disregarding multiples of ????: 0 ≤ ???? ≤ ???? Exactly definition of unit ball: 0,1 × 0,2???? × 0,???? ] Box can be altered just like 0,1 × 0,2????] o Ex. Evaluate the Jacobian of ???? ????,????,???? = ???? sin???? cos???? ,???? sin???? sin???? ,???? cos???? . ) ???????? ???????? ???????? ???????? ???????? ???????? sin???? cos???? −???? sin???? sin???? ???? cos???? sin???? |???????? ???????? ????????| = |sin???? sin???? ???? sin???? cos???? ???? cos???? cos???? | |???????? ???????? ????????| ???????? ???????? ???????? cos???? 0 −???? sin???? ???????? ???????? ???????? −???? sin???? sin???? ???? cos???? sin???? sin???? cos???? −???? sin???? sin???? = cos???? | ???? sin???? cos???? ???? cos???? cos???? | − 0 − ???? sin???? sin???? sin???? ???? sin???? cos???? | = ⋯ calculations = −???? sin???? 2 2 Recall absolutel value bars: ???? sin???? because ???? > 0 and sin???? ≥ 0 0 ≤ ???? ≤ ???? ) o Ex. Calculate the volume of a ball ???? of radius ???? using a change of variables to spherical coordinates. Pre-image under ???? is box 0,???? × 0,2???? × 0,????[ ] ???????????? = ∭ 1???????????????????????? ???????? 2???? ???? = ∫ ∫ ∫ 1 ???? sin???? ???????? ???????? ???????? 0 0 0 = ∫ ∫ 2????−???? cos???? | ???????? ???????? 0 0 0 = ∫ ∫ 2????−???? cos???? + ???? cos0???????? ???????? 0 0 = ∫ ∫ 2???????? + ???? ???????? ???????? 0 0 = 2 ∫ ???????? ????|2???? ???????? 0 0 = 4???? ∫???? ???? ???????? 0 1 3 ???? = 4???? ( 3 )| 4 0 = ???????? 3 3 M312 Section 6.3 Notes- Applications of Change of Variables (Centers of Mass) 2-3-16 How do we represent objects with mass mathematically? o We will talk about ℝ here; analogous to ℝ 2 3 o (1) Specify region of space ???? it occupies (in ℝ = wire, ℝ = sheet, ℝ = object) Not enough to simply specify total mass of object- need to be able to represent density at any point ???? o (2) Density function ????:ℝ → ℝ- given a point, conveys density of object near that point Explicit density function will be given Divide object into boxes, find mass withinibox, divide by volume oi box (gives approximation) Let limit of this notion be density of object at a point (let volume of cubes → 0) Restated: Suppose an object occupies a region ????, and ???? ???? ????; how do we compute the density of the object at ????? ???????????????? ???????? ???????????????? ???????? ???????????? ???????????????? ???????????????????????? ???? o As stated above, ???? ???? ≈ ???????????????????????? ???????? ???????????????? ???????? ???????????? o As radius of ball/box → 0, we get exact density at ???? Given an object with mass: o Total mass of object = scalar o Center of mass of object = point Computing the Mass of an Object o With object described by ???? and ????, to approximate mass: Partition ???? into boxes, add up mass of each box contained in ???? ( ) ???????????????? ???????? ???? For very small box ????, ???? ???????????? ???????? ???? ≈ ???????????? ????) So ???????????????? ???????? ???? = ???? ???????????? ???????? ???? ????(???? ???? ( )) more accurate as ???? gets smaller o Total mass ???? ≈ ∑ ???????????????????? ???????????????????? ????( ) ???????????????????????????????????? ???????? ???? ≈ ???? ???????????? ???????? ???? ????(???? ????( ) ) As partitions get finer, sum converges∬t???? ???? ???????????? ???????? ???? ???????? = ???????????????????? ???????????????? ???? ( ) ∴ ???????????????????? ???????????????? = ∬???? ???? ???????? or∭ ???? ???? ???????? according to which dimension you work in Ex. Consider an object that occupies the region 1,2 × 1,2 × 1,2 = 1,2 . Suppose the ( ) ( ) ???? density is given by ???? ???? = 1 + ???? ???? ????. Compute the mass of the object. o ???????????????? = ∭ 3(1 + ???? ???? ???????????????????????????? [1,2 = ∫ ∫ ∫ 2(1 + ???? ???? ???????????????????????????? 1 21 1 2 2 = (∫1 ???? ????????)( ∫1 ????????????)( 1 1 + ????????????) recall how whole integrals can be separated by pulling out everything constant in current variable of integration ????2 1 22 1 22 = ???? 1 )(2???? |)(???? + ????2 | ) 1 1 = ???? − ???? () 1(4 − 1 )( 2 + 2 − (1 + ))1 2 2 = ???? − ???? ( )(4 − ) 3 2 2 = (6 − ) ???? − ???? ) 4 15(???? − ???? ) 4 Centers of Mass o Suppose we have finitely many particles in ℝ labeled 1 through ???? with masses ???? ,…,???? ???? ℝ and ???? 1 ???? positions 1 ,…,???? ???? ℝ o Average position = 1(????1+ ⋯+ ???? ????) Center of mass not necessarily here ???? o Center of mass (COM) is weighted average of1???? through ???? depending on proportion of total mass each particle takes up ???? Proportion of mass for ???? : ???? ????1+⋯+???? ???? For finitely many points, ???????????? = ( ????1 )???? + ( ???? 2 )???? + ⋯+ ( ???? ???? )???? ???? 1⋯+???? ???? 1 ???? 1⋯+???? ???? 2 ???? 1⋯+???? ???? ???? Notice that coefficients add to 1; truly an average o For object O (infinitely many points) given by ???? and ????: ???????????????? ???? ) Partition ????: ???????????? ≈ ∑ ???????????????????? ???? (???????????? ???????? ???? (???????????????????? ???????????????? ???????????????????????????????????? ???????? ???? Can pull total mass out and rearrange with clever addition: 1 ???????????????? ????) ???????????? ≈ ???????????????????? ???????????????? ???????????????????? ???????????????? ???????? ???? ( ???????????????? ????)) (???????????? ????( )) As partitions get finer: ???????????????? (????) ???????????????? (????)→ ???? ???????????? ???????? ???? = density of box B ???????????????????? ???????????????? → ???????????????? (????) ∬ ????∙???? ???? ???????? Whole thing becomes Riemann Sum ???? ∬???? ???? ???? ???????? Numerator is vector-valued integrand (makes sense- COM is point): ∬???? ????∙???? ???? ???????? ∬????(????,???? ∙???? ????,???? ???????????????? ∬ ????????∙???? ????,???? ????????????∬,????????∙???? ????,???? ????????????????) = = ∬???? ???? ???? ???????? ∬???? ???? ????,???? ???????????????? ∬???? ???? ????,???? ???????????????? Directly analogous in 3 dimensions: (∬???? ????∙???? ????,????,???? ????????????????∬????,????∙???? ????,????,???? ????????????????∬????,????∙????(????,????,????)????????????????????????) ???????????? = ∬???? ???? ????,????,???? ???????????????????????? o Ex. Calculate the center of mass of 0,1if ???? ????,???? = ???? ????+????. Intuitively, density increases along diagonal going up and right; COM should be closer to 1,1 than origin 1 1 ∬ [ ]2???? ∙ ????(????,????)???????????????? = ∫ ∫ ???? ???? ????+????) ???????????????? 0,1 1 0 0 = ∫ (???? − 1 ????????+???? 0 ???????? integration by parts (?) 01 = ∫ 0 − −???? ???????? 0???? 1 = ???? |0 = ???? − 1 ∬ 2???? ∙ ????(????,????)???????????????? = ∬ 2???? ∙ ????(????,????)???????????????? because domain is same and ???? and ???? are [0,1 0,1 symmetric 1 1 ????+???? Total mass = ∫0∫ 0 ???? ???????????????? 1 ???? 1 ???? = 0 ???? ???????? ∫0 ???? ???????? = ???? − 1 )2 ( ) ???????????? = ????−1,????−1 = ( 1 , 1 ) ≈ 0.58,0.58 ) ????−1)2 ????−1 ????−1 o Uniform density = ???? constant ????(∬???? ????????????????????,∬???? ????????????????) ???????????? = ????∬???? 1???????????????? (∬ ????????????????????∬ ????????????????????) = ???? ???? ???????????????? (????) o Ex. Find the center of mass of the right triangle ???? with vertices 0,0 , ????,0 , and (0,????) and uniform density. both type I and type II region 1 ???????????????? ???? = ????????2 ???? ∬ ???????????????????? = ∫ ∫ −????????+???? ???????????????????? ???? 0 0 ???? − ????+???? = ∫ ???????? |0???? ???????? ???? ???? = ∫ ????(− ???? + ????)???????? ???? −???? ???? = ∫ ???? + ???????????????? 0 ???? ???? −???? 3 ???? 2 = 3???? ???? + 2 | 0 −???? 3 ???? 2 = 3???? ???? + 2 −???? ???? 1 = ???? + ???? = ???? ???? 2 3 2 6 ∬ ???????????????????? = ∬ ???????????????????? because ???? is both kinds of regions ???? 1 1 ???? (6???? ????6 ???? ????) 1 1 ???????????? = 1 = ( 6, 6) 2???????? M312 Exam 1 Information 2-10-16 Free Response Topics (slightly tentative)- 6 questions, choose 3 total over both class periods o Inspired by book problems sent out over email, not necessarily directly related o Not harder than book problems given o Breakdown: 2 problems involving objects with mass 1 problem emphasizing spherical coordinates 1 or 2 problems of the form “given a transformation and a domain, find the image or pre-image.” OR “prove that ???? is 1-1 on ????” (likely not linear, not polar; something newer (?)) 1 or 2 “regular” integration problems involving change of variables 10 multiple choice o Similar in distribution and breakdown to ones sent out over email

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