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Problems are grouped by topic, and each section includes theorems, definitions, worked examples, and practice questions. Solutions are provided primarily for verification and deeper understanding, encouraging independent completion of all problems and targeted mastery of limits, differentiation, integration, and key vector theorems.",{"@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":46,"@type":70,"position":76},"https://docshare.wps.com/template/paper-templates/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/study-guide-of-calculus-iii-key-topics-and-problems/573140/",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/study-guide-of-calculus-iii-key-topics-and-problems/573140.png","ImageObject",442,249,{"name":88,"@type":89},"RuangKosong","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-15","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":9},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"What major topics are covered in Calculus III in this study guide?","Question",{"text":108,"@type":109},"The guide covers partial derivatives, multiple integrals, and vector calculus, organized into thematic topics and problems aligned with the Tests and the Final.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"How should problems be used relative to the provided solutions?",{"text":113,"@type":109},"Complete and understand each problem independently; use the solutions only to verify answers and enhance understanding.",{"name":115,"@type":106,"acceptedAnswer":116},"How is the guide structured for practice?",{"text":117,"@type":109},"Problems are collected by topic, and the guide closely mirrors the type and difficulty of problems on the Tests and the Final.","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},573140,1792057071,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":45,"category_name":46,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":9,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":141},962090883219,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","STUDY GUIDE OF CALCULUS III  \nXIAOLONG HAN  \nAbstract. The three main subjects in Calculus III are partial derivatives, multiple integrals, and vector calculus. This Study Guide covers the key topics and problems featured in the Tests and the Final.  \n(I) This Guide presents a collection of problems organized by topic. These problems closely reflect those on the Tests and the Final in type and difficulty.  \n(II) Complete and understand all problems independently. Solutions are provided, but use them only to verify answers and enhance understanding.  \nContents  \nTopic 1 . Lines, planes, and quadratic surfaces 2  \nTopic 1.A. Lines 2  \nTopic 1.B. Planes 2  \nTopic 1.C. Quadratic surfaces 3  \nTopic 2 . Differentiation 4  \nTopic 2.A. Limits 4  \nTopic 2.B. Partial derivatives and chain rule 5  \nTopic 2.C. Directional derivatives and gradients 6  \nTopic 2.D. Tangent planes 7  \nTopic 2.E. Derivative tests 8  \nTopic 2.F. Lagrange multiplier 9  \nTopic 3 . Integration 9  \nTopic 3.A. Double integrals 10  \nTopic 3.B. Triple integrals 11  \nTopic 4 . Vector calculus 15  \nTopic 4.A. Conservative vector fields 16  \nTopic 4.B. Green’s theorem 18  \nTopic 4.C. Surface integrals 20  \nTopic 4.D. Stokes’ theorem 21  \nTopic 4.E. Divergence theorem 23  \n2 XIAOLONG HAN  \nTopic 1 . Lines, planes, and quadratic surfaces  \nTopic 1.A. Lines.  \nTheorem (Lines) . A line that passes through a point P0 = (x0 , y0 , z0 ) in the direction of the vector v = ha, b, ci has the vector equation  \nhx, y, zi = hx0 , y0 , z0 i + tha, b, ci,  \nand equivalently the parametric equations  \nx = x0 + at, y = y0 + bt, z = z0 + ct.  \nProblem 1 . Find the equation of the line that passes through the point (2 , 5 , −1) and is orthogonal to the plane 3x − 2y + z = 7 .  \nAnswer. The plane 3x − 2y + z = 7 has a normal vector h3 , −2 , 1i. The line orthogonal to this plane is in the direction of this vector. Hence, it has the vector equation  \nhx, y, zi = h2 , 5 , −1i + th3 , −2 , 1i,  \nand equivalently the parametric equations  \nx = 2 + 3t, y = 5 − 2t, z = −1 + t.  \nTopic 1.B. Planes.  \nTheorem (Planes) . A plane that passes through a point P0 = (x0 , y0 , z0 ) with a normal vector n = ha, b, ci has the equation  \nha, b, ci · hx − x0 , y − y0 , z − z0 i = 0 ,  \nthat is,  \na (x − x0 ) + b (y − y0 ) + c (z − z0 ) = 0 .  \nProblem 2 . Find an equation of the plane that contains the point (0 , 1 , 2) and the line r(t) = h2 − t, 1 + t, 1 + 2ti.  \nAnswer. The line r(t) = h2 − t, 1 + t, 1 + 2ti is in the direction of the vector u = h−1 , 1 , 2i. A plane that contains this line has its normal vector n orthogonal to u.  \nA plane that contains the line r(t) = h2 − t, 1 + t, 1 + 2ti contains the point (2 , 1 , 1) (by setting t = 0) . Hence, this plane contains the line passing through (2 , 1 , 1) and (0 , 1 , 2), which is in the direction of the vector v = h2 − 0 , 1 − 1 , 1 − 2i = h2 , 0 , −1i. The plane has its normal vector n orthogonal to v.  \nThe plane has a normal vector  \nn = u × v = h−1 , 1 , 2i × h2 , 0 , −1i = h−1 , 3 , −2i.  \nSince the plane contains the point (0 , 1 , 2), it has the equation  \n−1(x − 0) + 3(y − 1) − 2(z − 2) = 0 .  \nHence,  \n−x + 3y − 2z = −1 .  \nThat is,  \nx − 3y + 2z = 1 .  \nSTUDY GUIDE OF CALCULUS III 3  \nTopic 1.C. Quadratic surfaces.  \nDefinition (Intercepts and traces) .  \n• Set y = z = 0 . The solutions are the x-intercepts of the surface, i.e. , the intersection of the surface and the x-axis. The y-intercepts and z-intercepts are obtained similarly.  \n• Set z = 0 . The solution is the xy-trace of the surface, i.e. , the intersection of the surface and the xy-plane. The yz-trace and xz-trace are obtained similarly.  \nProblem 3 . Sketch the surface x2 − y 2 + z 2 = 4 .  \nAnswer. The yz-trace, −y2 + z 2 = 4, is a hyperbola, which intersects the z-axis at ±2 and does not intersect the y-axis.  \nThe xz-trace, x2 + z 2 = 4, is a circle with center at the origin and radius 2.  \nThe surface is a hyperboloid of one sheet along the y-axis. See the sketch below.  \n4 XIAOLONG ","cbCaimkZ7mqn0VIK","https://ap.wps.com/l/cbCaimkZ7mqn0VIK","pdf",751828,24,"English","# Topic 1. Lines, planes, and quadratic surfaces\n## Topic 1.A. Lines\n## Topic 1.B. Planes\n## Topic 1.C. Quadratic surfaces\n# Topic 2. Differentiation\n## Topic 2.A. Limits\n## Topic 2.B. Partial derivatives and chain rule\n# Topic 3. Integration\n## Topic 3.A. Double integrals\n## Topic 3.B. Triple integrals\n# Topic 4. Vector calculus\n## Topic 4.A. Conservative vector fields\n## Topic 4.B. Green’s theorem\n## Topic 4.C. Surface integrals\n## Topic 4.D. Stokes’ theorem\n## Topic 4.E. Divergence theorem","[{\"question\":\"What major topics are covered in Calculus III in this study guide?\",\"answer\":\"The guide covers partial derivatives, multiple integrals, and vector calculus, organized into thematic topics and problems aligned with the Tests and the Final.\"},{\"question\":\"How should problems be used relative to the provided solutions?\",\"answer\":\"Complete and understand each problem independently; use the solutions only to verify answers and enhance understanding.\"},{\"question\":\"How is the guide structured for practice?\",\"answer\":\"Problems are collected by topic, and the guide closely mirrors the type and difficulty of problems on the Tests and the Final.\"}]","STUDY GUIDE OF CALCULUS III - Key Topics and Problems | PDF",1791933167,8]