[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573137-105":53,"doc-detail-573137-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","ma2104-multivariable-calculus-basic-vectors","MA2104 - Multivariable Calculus - Basic Vectors","","Multivariable Calculus notes focused on vectors and key operations, covering dot product, component (signed scalar), projections, and orthogonality. It derives cross product formulas and angle relationships, then extends to the scalar triple product for signed volume and plane equations. The content also includes vector differentiation rules, arc length, and 2D/3D level sets, contour and contour surface concepts. It further covers limits, continuity, tangent plane and differentiability, and chain/implicit differentiation techniques.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/ma2104-multivariable-calculus-basic-vectors/573137/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/ma2104-multivariable-calculus-basic-vectors/573137.png","ImageObject",442,249,{"name":88,"@type":89},"PakDamar76","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":79},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What are the main properties of the dot product in these notes?","Question",{"text":108,"@type":109},"The notes list commutativity and distributive properties, along with angle and orthogonality relationships, including when the dot product equals zero.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How are projections and components (signed scalar) defined?",{"text":113,"@type":109},"Components are expressed using magnitudes and cosine of the angle, and projection formulas use the same scalar component to obtain the vector projection direction.",{"name":115,"@type":106,"acceptedAnswer":116},"What conditions classify a point as a critical point, local max, local min, or saddle point?",{"text":117,"@type":109},"A critical point requires both partial derivatives to be zero. Local maxima/minima require nearby points to have function values not exceeding/not less than the point, while saddle points have nearby points where the function is both greater and less.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},573137,1792022861,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":79,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":73,"language":135,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":61,"update_tm":139,"read_time":9},962090893057,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","MA2104  \nMultivariable Calculus  \nBasic Vectors  \n❼ Thm 1 : kcuk = jcj kuk  \n❼ Thm 2: (unit vector in direction of a) = k~~a~~ak  \n❼ Thm 3 [Dot product properties] :  \na 􀀁 b = b 􀀁 a (da) 􀀁 b = d (a 􀀁 b) = a 􀀁 (db) a 􀀁 (b + c) = a 􀀁 b + a 􀀁 c 0 􀀁 a = 0  \n(a + b) 􀀁 c = a 􀀁 c + b 􀀁 c a 􀀁 a = kak2  \n❼ Thm 4 [Dot product & angle] : a 􀀁 b = kak kbk cos 􀀒❼ Thm 5 [Orthogonality] : a ? b () a 􀀁 b = 0  \n❼ Component (signed scalar): compa b = kbk cos 􀀒 = ~~a~~ka􀀁~~ b~~k ❼ Projection (vector): proja b = compa b 􀀂 k~~a~~ak = ~~a~~a􀀁􀀁~~b~~aa  \n􀀌 i j k 􀀌❼ Cross product: ha1 ; a2 ; a3 i 􀀂 hb1 ; b2 ; b3 i := 􀀌 a 1 a2 a3 􀀌 =  \n􀀌 b 1 b2 b3 􀀌 ha2 b3 􀀀 a3 b2 ; a3 b 1 􀀀 a 1 b3 ; a 1 b2 􀀀 a2 b 1 i  \n❼ Thm 6: (a 􀀂 b) ? a and (a 􀀂 b) ? b  \n❼ Thm 7 [Cross prod. & angle] : ka 􀀂 bk = kak kbk sin 􀀒  \n❼ Thm 8 [Cross product properties] :  \na 􀀂 b = 􀀀 (b 􀀂 a) (a + b) 􀀂 c = a 􀀂 c + b 􀀂 ca 􀀂 (b + c) = a 􀀂 b + a 􀀂 c (da)􀀂b = d (a 􀀂 b) = a􀀂 (db)  \n❼ Scalar triple product (= signed vol. of parallelepiped):  \n􀀌 a 1 a2 a3 􀀌 a 􀀁 (b 􀀂 c) := 􀀌 b 1 b2 b3 􀀌 =  \n􀀌 c 1 c2 c3 􀀌  \na 1 (b2 c3 􀀀 b3 c2 ) ; a2 (b3 c 1 􀀀 b1 c3 ) ; a3 (b1 c2 􀀀 b2 c 1 )  \n❼ Thm 10 & 11 [Plane] :  \nn 􀀁 r = n 􀀁 r0 () ax + by + cz = ax0 + by0 + cz0 = d  \n❼ Thm 13 [Derivative properties for vectors] : ~~d~~dt (r (t) + s (t)) = r0 (t) + s0 (t)  \n~~d~~dt (c r (t)) = c r0 (t)  \n~~d~~dt (f (t) r (t)) = f 0 (t) r (t) + f (t) r0 (t)  \n~~d~~dt (r (t) 􀀁 s (t)) = r0 (t) 􀀁 s (t) + r (t) 􀀁 s0 (t)  \n~~d~~dt (r (t) 􀀂 s (t)) = r0 (t) 􀀂 s (t) + r (t) 􀀂 s0 (t)  \n❼ Thm 14 [Arc length] : (length from a to b) = Rba kr0 (t)k dt  \n❼ Vector rotation: 90 ➦ anticlockwise: hx; yi ! h􀀀y; xi 90 ➦ clockwise: hx; yi ! hy; 􀀀xi  \nSurfaces  \n❼ Level curve of f(x; y) = horizontal trace (for functions in two vars) = 2-D graph of f (x; y) = k for some constant k Contour plot = numerous level curves on the same graph  \n❼ Level surface of f(x; y; z) = 3-D graph of f (x; y; z) = k for some constant k.  \nQuadric surfaces  \n❼ Cylinder = in􀀌nite prism  \n❼ Elliptic paraboloid: ~~x~~a2~~2~~ + yb2~~2~~ = ~~z~~c   \n❼ Hyperbolic paraboloid: ~~x~~a2~~2~~ 􀀀 yb2~~2~~ = ~~z~~c   \n❼ Ellipsoid: ~~x~~a2~~2~~ + yb2~~2~~ + ~~z~~c2~~2~~ = 1   \n❼ Elliptic cone: ~~x~~a2~~2~~ + yb2~~2~~ 􀀀 ~~z~~c2~~2~~ = 0   \n❼ Hyperboloid : x2 + y2 􀀀 z2 = 1 of one sheet a2 b2 c2  \n❼ Hofyptwerobsohloeidets : ~~x~~a2~~2~~ + yb2~~2~~ 􀀀 ~~z~~c2~~2~~ = 􀀀1   \nLimits  \n❼ Limit: lim f (x; y) = L  \n(x;y)! (a;b)  \ni􀀋 for any 􀀏 > 0 there exists 􀀎 > 0 such that  \nj f(x; y) 􀀀 Lj \u003C 􀀏 whenever 0 \u003C q (x 􀀀 a)2 + (y 􀀀 b)2 \u003C 􀀎  \n❼ Thm 15: To show limit does not exist, take the limit via two di􀀋erent paths that have di􀀋erent limits  \n❼ Thm 16 & 17 [Limit theorems] : Limits may be taken into addition, subtraction, multiplication, division  \n❼ Thm 18 [Squeeze theorem] :  \nIf j f(x; y) 􀀀 Lj 􀀔 g(x; y) 8 (x; y) close to (a; b) and lim g (x; y) = 0  \n(x;y)! (a;b)  \nthen lim f (x; y) = L  \n(x;y)! (a;b)  \n❼ Continuity: f is continuous at (a; b)() lim f (x; y) = f(a; b)  \n(x;y)! (a;b)  \ni.e. the limit exists and the f is valid at (a; b)  \n❼ Thm 20 & 21 [Continuity theorems] :  \nIf two functions are continuous (at (a; b)), then their sum, di􀀋erence, product, quotient, and composition are continuous too (quotient requires denominator  0)  \n❼ All polynomials, trigonometric, exponential, and rational functions are continuous  \nPartial Derivatives  \n❼ Thm 2 [Clairaut's theorem]: If fxy and fyx are both continuous on disk containing (a; b) then fxy (a; b) = fyx (a; b)  \n❼ Thm 3 [Tangent plane eqn] :  \nGiven surface z = f (x; y) with point (a; b):  \n-normal vector: hfx (a; b); fy (a; b); 􀀀1i  \n-tangent plane: z = f (a; b) + fx (a; b)(x􀀀a) + fy (a; b)(y 􀀀b)  \n❼ Multivariable di􀀋erentiability: z = f (x; y) is di􀀋erentiable at (a; b) if  \n4z = fx (a; b)4x + fy (a; b)4y + 􀀏14x + 􀀏24y  \n(with vanishing 􀀏 1 and 􀀏2 ) i. e. zooming in to (a; b) will make surface approximate tangent plane  \n❼ fx & fy are continuous at (a; b) =) f is di􀀋.able at (a; b)  \n❼ f is di􀀋erentiable at (a; b) =) f is continuous at (a; b)  \nDi􀀋er","cbCaiczgsrqX2asn","https://ap.wps.com/l/cbCaiczgsrqX2asn","pdf",2714520,"English","# Basic Vectors\n## Dot product properties\n## Cross product and angles\n## Scalar triple product and plane equations\n## Vector differentiation and arc length\n# Surfaces and Limits\n## Level curves and level surfaces\n## Quadric surfaces\n## Limits and continuity\n# Partial Derivatives and Differentiation\n## Clairaut's theorem and tangent plane\n## Differentiability and continuity\n## Chain rule and implicit differentiation\n# Gradient, Critical Points\n## Directional derivatives and gradients\n## Level set normality\n## Critical points and extrema","[{\"question\":\"What are the main properties of the dot product in these notes?\",\"answer\":\"The notes list commutativity and distributive properties, along with angle and orthogonality relationships, including when the dot product equals zero.\"},{\"question\":\"How are projections and components (signed scalar) defined?\",\"answer\":\"Components are expressed using magnitudes and cosine of the angle, and projection formulas use the same scalar component to obtain the vector projection direction.\"},{\"question\":\"What conditions classify a point as a critical point, local max, local min, or saddle point?\",\"answer\":\"A critical point requires both partial derivatives to be zero. Local maxima/minima require nearby points to have function values not exceeding/not less than the point, while saddle points have nearby points where the function is both greater and less.\"}]","MA2104 - Multivariable Calculus - Basic Vectors | PDF",1791933165]