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Calculus with Multiple Variables Essential Skills Workbook Includes Vector Calculus and Full Solutions Chris McMullen, Ph.D.Copyright © 2021, 2023 Chris McMullen, Ph.D.www.improveyourmathfluency.com www.monkeyphysicsblog.wordpress.com www.chrismcmullen.com Zishka Publishing All rights reserved.
ISBN: 978-1-941691-37-3
Textbooks > Math > Calculus Study Guides > Workbooks > Math Education Math > Calculus 2 / 10
Contents Introduction iv
- Partial Derivatives 5
- The Chain Rule with Multiple Variables 11
- Extreme Values with Multiple Variables 15
- Vectors 19
- Scalar and Vector Products 29
- Polar Coordinates 35
- Spherical Coordinates 41
- Cylindrical Coordinates 47
- The Gradient 53
10 The Divergence 57 11 The Curl 63 12 Normal and Tangent Vectors 67 13 Line Integrals 77 14 Surface and Volume Integrals 87 15 Center of Mass and Moment of Inertia 101 Solutions111 3 / 10
Introduction This workbook is designed to help practice a variety of practical calculus skills that involve multiple variables, including vector calculus. Each chapter focuses on one main topic, like how to apply the gradient operator or how to perform a center of mass integral.Prerequisites: The student should already be fluent in derivatives and integrals of polynomials, basic trig functions, exponentials, and logarithms.Every chapter begins with a concise explanation of pertinent concepts, followed by a few examples. Every example is fully solved step-by-step with explanations. The examples should serve as a handy guide for how to solve the practice exercises. Every exercise is fully solved at the back of the book.A variety of multivariable calculus skills are covered. This workbook begins with partial differentiation and finishes with various multivariable integrals. Students will
learn:
• how to take a partial derivative.• how to find the minimum and maximum values of a function of two variables.• basic properties of vectors, including the scalar and vector product.• essential properties of polar, spherical, and cylindrical coordinates.• how to apply the gradient, divergence, and curl operators.• how to integrate an expression over a path.• how to perform double and triple integrals.• how to perform surface and volume integrals.• how to perform center of mass integrals.• how to perform moment of inertia integrals.May you (or your students) find this workbook useful and become more fluent with these essential multivariable calculus skills. 4 / 10
- Partial Derivatives
When taking a partial derivative of a function with respect to one variable, treat the other independent variables as if they are constants. The symbol d is used (instead of the letter d) to represent a partial derivative. For example, represents a partial derivative of the function f with respect to the variable x. (In contrast, ^2 represents a total derivative. See Chapter 2.) Example. Given z = 4%3y2, find and .When finding ^|, treat the independent variable y as if it were a constant.= 4%3y2 = 4y2-— x3 = 4y2(3%2) = 12%2y2 ox ox ox Similarly, when finding , treat the independent variable x as if it were a constant.— = — 4%3v2 = 4x3 — y2 = 4%3(2v) = 8x3y dydy dy . , df , df Example. Given f = 3% sin t, find and —.When finding treat the independent variable t as if it were a constant.— = = — 3x sin t = 3 sin t — x = 3 sin t (1) = 3 sin t dx dx dx k J Similarly, when finding , treat the independent variable x as if it were a constant.— = = — 3x sin t = 3x — sin t = 3%(cos t) = 3x cos t ut ut ut To find a second partial derivative, take one partial derivative at a time. For example, ^2 can be found as Q0. Note that a “mixed” partial derivative is possible, such as ^-4~, which means (^2). For most common standard functions, ^-4- is equal to 4~4~ dxdy dxXdyJ ' dxdy dydx except near discontinuities, in accordance with Clairaut’s theorem.
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Chapter 1 - Partial Derivatives Example. Given z = 6%4y2, find —.First find , treating the independent variable y as if it were a constant.— = — 6%4y2 = 6y2 — x4 = 6y2(4%3) = 24x3y2 dx dx ox Now find (^|), treating the independent variable y as if it were a constant.—= = — f—) = — 24%3y2 = 24v2 —%3 = 24v2(3%2 ) = 72%2v2 ox2 ox \dx/ ox ox Example. Given f = x3y + %2v2, find ^-4- and .r 1 J J dydx dxdy When finding |£, treat the independent variable y as if it were a constant.—- = — (x3y + %2y2) = y — x3 + y2 — x2 = y(3%2) + y2(2x) = 3x2y + 2xy2 ox ox ox ox Similarly, when finding treat the independent variable x as if it were a constant.—- = — (x3y + %2y2) = x3 — y + x2 — y2 = %3(1) + x2 (2y) = x3 + 2x2y dy dy dy dy Now find the mixed second derivatives.d2f d (df\ d 2 . 2d d 2 w = ayfc)=~(3xy+ 2xy) = 3x -^y+ 2*^ = 3%2 (1) + 2x(2y) = 3%2 + 4xy fL = *(£) = *(X3 + 2X2V)= p + 2*2 dxdy dx \dy/ dx dx dx = 3%2 + 2y(2x) = 3%2 + 4xy Interpretation of partial derivatives: The equation z = /(%, y) represents a surface S.The point (a, b, c) lies on S if c = f(a, h). The intersection of the vertical plane x = a and the surface S is the curve C1 (which is called the trace of S in the plane x = d) and the intersection of the vertical plane y = b and the surface S is the curve C2 (which is called the trace of S in the plane y = h). The partial derivatives and evaluated at (a, h) give the slopes of the tangent lines of the traces C1 and C2 at the point (a, b, c) in the planes x = a and y = b.
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Calculus with Multiple Variables Essential Skills Workbook Chapter 1 Exercises - Part A Directions: Perform each partial derivative with respect to the indicated variable.O Given z = , find and .
- , dx dy
- Given z = -, find and .
- dx dy
y ox dy O Given f = xJy, find and ^4
Q Given w = sin t cos u, find and .❖ Check your answers at the back of the book.
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Chapter 1 - Partial Derivatives Chapter 1 Exercises - Part B Directions: Perform each partial derivative with respect to the indicated variable.& Given g = ey ln x, find and |^., dx dy @ Given z = x4 + 2%2y2, find and .' dx dy
- Given u = jp2 — q2, find and |^.& Given h = ln(t2 + Lu), find and |^.
at au ❖ Check your answers at the back of the book.
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Calculus with Multiple Variables Essential Skills Workbook Chapter 1 Exercises - Part C Directions: Perform each partial derivative with respect to the indicated variable.Q Given f = ^, find and y^.^2 J y2 dx2 dy2 © Given w = t2 sin u, find ^- and ^-.❖ Check your answers at the back of the book.
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Chapter 1 - Partial Derivatives Chapter 1 Exercises - Part D Directions: Perform each partial derivative with respect to the indicated variables.^ Given z = , find and .
- , dydx dxdy
® Given z = exy, find and ., dydx dxdy Check your answers at the back of the book.10