[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-573139-105":53,"doc-detail-573139-en":120},{"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":113,"head_meta":115,"extra_data":117,"updated_unix":119},105,"en","cheat-sheet-for-midterm-3-m252-calculus-iii-fall-2022","Cheat Sheet for Midterm #3 - M252 - Calculus III - Fall 2022","","Cheat Sheet for Midterm #3 in M252 Calculus III (Fall 2022) summarizes key multivariable calculus formulas without requiring memorization. It emphasizes understanding how to apply each result, including correct variable choice for partial derivatives and integrals. The sheet covers gradient and direction of steepest ascent, multiple coordinate systems (polar, cylindrical, spherical), triple integral types, surface area and volume, mass via density, change of variables using the Jacobian, and both scalar and vector line integrals. It also includes conservative field criteria and Green’s theorem with orientation rules.",{"@graph":63,"@context":112},[64,80,95],{"@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":36,"@type":70,"position":76},"https://docshare.wps.com/template/forms/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/cheat-sheet-for-midterm-3-m252-calculus-iii-fall-2022/573139/",4,{"url":78,"name":59,"@type":81,"author":82,"headline":59,"publisher":85,"fileFormat":88,"inLanguage":57,"description":61,"dateModified":89,"datePublished":89,"encodingFormat":88,"isAccessibleForFree":90,"interactionStatistic":91},"DigitalDocument",{"name":83,"@type":84},"OmBimo","Person",{"url":68,"name":86,"@type":87},"DocShare","Organization","application/pdf","2026-10-13",true,{"@type":92,"interactionType":93,"userInteractionCount":4},"InteractionCounter",{"@type":94},"ViewAction",{"@type":96,"mainEntity":97},"FAQPage",[98,104,108],{"name":99,"@type":100,"acceptedAnswer":101},"Does the cheat sheet require memorizing all formulas?","Question",{"text":102,"@type":103},"No. The sheet states that there is no need to memorize the formulas, but it stresses understanding how to apply all of them.","Answer",{"name":105,"@type":100,"acceptedAnswer":106},"What caution is given when taking derivatives or integrals of functions of several variables?",{"text":107,"@type":103},"It warns to be careful about which variable you differentiate (or integrate) with respect to, reminding readers to track “Integral (or derivative) with respect to what.”",{"name":109,"@type":100,"acceptedAnswer":110},"How does Green’s theorem relate to orientation of the curve?",{"text":111,"@type":103},"Green’s theorem is given with the requirement that the path must be oriented anti-clockwise, and if punching holes, inner paths must be oriented clockwise.","https://schema.org",{"og:url":78,"og:type":114,"og:title":59,"og:site_name":86,"og:description":61},"article",{"robots":116,"canonical":78},"index,follow",{"doc_id":118,"site_id":56},573139,1791933166,{"code":4,"msg":5,"data":121},{"doc_id":118,"user_id":122,"nickname":83,"user_avatar":123,"doc_module":9,"category_id":35,"category_name":36,"doc_title":59,"doc_description":61,"doc_content":124,"file_id":125,"file_url":126,"file_type":127,"file_size":128,"view_count":4,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":9,"language":129,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":130,"faqs":131,"seo_title":132,"seo_description":61,"update_tm":119,"read_time":4},962090893581,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Cheat Sheet for Midterm\\#3-M252-Calculus III-Fall 2022  \nThis cheat sheet will be included in the midterm. You do NOT have to memorize these formulas. However, make sure that you understand and know how to apply ALL of them!!!  \n• If a constant is not de􀀌ned (like an α or any other symbol), do NOT assign a value for it. It is a 􀀌xed scalar (or vector) and you should just leave it as it is.  \n• When dealing with functions of several variables, be careful about which variable you are doing the derivative (or integral) with respect to! [Remember the: “Integral (or derivative) with respect to what?”]  \n      −→        \n• Gradient: Nabla operator = ∇ = h ∂∂x , ∂∂y , ∂∂z i ||| gradient(f) = ∇f(x, y, z) = h ∂∂x , ∂∂y , ∂∂z if = hfx , fy , fzi.  \n• Direction of steepest ASCENT is given by gradient ∇f(x, y, z) ||| Gradient vector ⊥ level curves.  \n• Polar: 􀀃D f (x, y) dA = 􀀃D f (r cos(θ), r sin(θ)) r dr dθ . ||| r2 = x2 + y 2 , x = r cos(θ), y = r sin(θ) .  \n• Cylindrical: 􀀅B f (x, y, z) dV = 􀀅B f (r cos(θ), r sin(θ), z) r dr dθ dz.  \nwith r2 = x2 + y 2 + z 2 , x = r cos(θ), y = r sin(θ), z = z.  \n• Spherical: 􀀅B f (x, y, z) dV = 􀀅B f (ρsin(φ)cos(θ),ρsin(φ)sin(θ),ρcos(φ)) ρ2 sin(φ) dρdθ dφ . with ρ2 = x2 + y 2 + z 2 , x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), z = ρcos(φ) .  \n• Type I, type II, and type III for triple integrals.  \n• Surface area: S = 􀀃D qf2x + f2y + 1 dA.  \n• Volume: V (B) =􀀅B dV  \n• Mass: M (B) =􀀅B ρ (x, y, z) dV where ρ = density.  \n• Change of variables (x, y, z) → (u, v, w): 􀀅 f (x, y, z) dV = 􀀅 f (u, v, w) |J| du dv dw where J = 􀀌 ((xu,,yv,,zw)) 􀀌 is the Jacobian of the transformation. [Similar for double integrals]  \n• Line integrals for scalar 􀀌elds:    \n◦ 2D: 􀀁C f (x, y) dS = 􀀁ta f (x(t), y (t)) p(dx/dt)2 + (dy/dt)2 dt ||| x = x (t) & y = y (t) for a ≤ t ≤ b  \n◦ 3D (and 2D): 􀀁C f (x, y, z) dS = 􀀁ta f (~r(t)) | ~r ′(t)| dt ||| ~r(t)=hx(t), y (t), z (t)i for a ≤ t ≤ b  \n• Line integrals for vector 􀀌elds: 􀀁C ~F · d~r = 􀀁C ~F · ~T dS = 􀀁ta ~F · ~r ′ dt  \n• Work: ~F(x, y, z) = hP(x, y, z), Q (x, y, z), R (x, y, z)i: W = 􀀁C ~F ·d~r = 􀀁ta ~F · ~r ′ dt = 􀀁 P dx+Qdy+Rdz −→  \n• Fundamental theorem: 􀀁C ∇f · d~r = f (~r(b)) − f (~r(a))  \n• A vector 􀀌eld is conservative if it comes from the gradient of a scalar 􀀌eld.  \n~  \n• F (x, y) = hP(x, y), Q (x, y)i conservative ⇔ Py = Qx [if on a simply-connected domain] .  \n• Green’s theorem (GT): 􀀋C P dx + Qdy = 􀀃D (Qx − Py) dA [path must be oriented anti-clockwise] .  \n• Punching holes: inner paths must be oriented clockwise [i.e. , opposite] . Remember: 􀀋−C = −􀀋C  \n• Area using GT: Area enclosed by C: A = 􀀋C xdy = 􀀋C −y dx = ~~1~~2 􀀋C xdy − y dx  \n• Make sure that you understand and know how to apply ALL the formulas given in the cheat sheet!","cbCaipHfpyLFoyCQ","https://ap.wps.com/l/cbCaipHfpyLFoyCQ","pdf",48041,"English","# Key Multivariable Formulas\n## Differentiation and Gradient Concepts\n## Coordinate Systems and Integrals\n## Line Integrals, Work, and Fundamental Theorem\n## Conservative Fields and Green's Theorem","[{\"question\":\"Does the cheat sheet require memorizing all formulas?\",\"answer\":\"No. The sheet states that there is no need to memorize the formulas, but it stresses understanding how to apply all of them.\"},{\"question\":\"What caution is given when taking derivatives or integrals of functions of several variables?\",\"answer\":\"It warns to be careful about which variable you differentiate (or integrate) with respect to, reminding readers to track “Integral (or derivative) with respect to what.”\"},{\"question\":\"How does Green’s theorem relate to orientation of the curve?\",\"answer\":\"Green’s theorem is given with the requirement that the path must be oriented anti-clockwise, and if punching holes, inner paths must be oriented clockwise.\"}]","Cheat Sheet for Midterm #3 - M252 - Calculus III - Fall 2022 | PDF"]