Calculus III Worksheets, Study notes of Vector Analysis

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Calculus III (2934, Fall 2019)

Worksheets

Kimball Martin

October 15, 2019

Worksheet 1: R^2 and R^3

1. What does R^2 mean? Can you give a precise definition? What about R^3? Rn?
2. What is the right-hand rule (la regolla della mano destra)?
3. Graph x = y in R^2.
4. Graph x = y in R^3.
5. Graph x^2 + y^2 = 1 in R^3.
6. Geometrically describe

(x, y, z) : 1 ≤ √x^2 + y^2 + z^2 ≤ 4

7. Write an equation for the sphere of radius 5 in R^3 centered at (1, 2 , 3).
8. Find the distance between the points (1, 2 , 3) and (2, 4 , 6) in R^3.
9. Graph x + y + z = 1.
10. Graph x^2 + y^2 = z in R^3.
11. Graph x^2 + y^2 = z^2 in R^3.
12. What are your strategies for graphing equations/inequalities in R^3?

Worksheet 3: Products of Vectors

1. What is the difference between scalar, dot and cross products? When do they make sense? What do they mean geometrically?
2. Do all of these products commute? E.g., is u · v = v · u always?
3. Do any of these products preserve length? E.g., is |u · v| = |u| · |v|?
4. Do any of these products have a cancellation property? E.g., does u · v = u · w imply v = w? (maybe assuming something is nonzero)
5. Use the dot product to compute the projections of u = 〈 2 , 1 〉 onto v = 〈 1 , 0 〉 and onto w = 〈 1 , − 1 〉. Can you interpret the projections in terms of work?
6. Is there a difference between the projection of a u onto v and the projection of u onto −v? What about the projection of v onto u?
7. Compute i × j and 〈 1 , 2 , 3 〉 × 〈 1 , 0 , − 1 〉. Sketch pictures.
8. Consider vectors a, b and c in R^3. Explain why a, b and c being coplanar means the scalar triple product a · (b × c) = 0. Is the converse also true?

Worksheet 4: Lines and Planes

1. Find vector, parametric and symmetric equations for the line through (1, 2 , −1) and the origin. Sketch.
2. Find vector equations for the line through (1, 2 , −1) which is parallel to the y-axis. Sketch. What about parametric or symmetric equations?
3. Describe the line in the previous problem as the intersection of 2 planes. Is there an easy way to find another pair of planes whose intersection is this line?
4. Find vector and linear equations for the plane containing (1, 0 , 0), (0, 1 , 0) and (0, 0 , 1). Sketch.
5. Find a normal vector for the plane in the last problem. Sketch.

Worksheet 6: Space curves

1. Sketch the curve C given by r(t) = 〈t, t^2 , t^3 〉, 0 ≤ t ≤ 1. Can you find another parametrization for C?
2. Let r(t) = cost ti + sint tj + (^1) t k for 0 < t < ∞. Find limt→∞ r(t) if it exists. Sketch the curve.
3. Find two curves from (0, 0 , 0) to (2, − 1 , −1). For each curve give vector and parametric equations, and sketch the curve.
4. Find two curves lying on the sphere S : x^2 + y^2 + z^2 = 1. For each curve give vector and parametric equations, and sketch the curve.
5. Let C be the intersection of x^2 + y^2 = 1 and y + z = 1. Find a parametric description for C.

Worksheet 7: Derivatives and Integrals of Vector Functions

For r(t) = 〈f (t), g(t), h(t)〉 a vector function, define the derivative r′(t) = 〈f ′(t), g′(t), h′(t)〉 and definite integral ∫^ ab r(t) dt = 〈∫^ ab f (t) dt, ∫^ ab g(t) dt, ∫^ ab h(t) dt〉 componentwise (when these quantities exist).

1. (a) Give a physical interpretation for r′(t) in terms of motion. (b) Give a physical interpretation for ∫^ ab r(t) in terms of motion.
2. Let r(t) = (cos t)i + (sin t)j + tk. Compute r′(0). Sketch the graph of r(t) and graph the vector r′(0) starting at the base point r(0).
3. Explain why r′(a) is a tangent vector to the curve given by r(t) at t = a (for general r(t) differentiable at t = a).
4. Compute the unit tangent vector at t = 0 for r(t) as in Problem 1. Why might it be useful to look at a unit tangent vector rather than just a tangent vector?
5. (a) Give an example of a vector function r(t) which is not continuous at t = a. (b) Give an example of a vector function r(t) which is continuous at t = a but not differentiable at t = a. (c) For each of the two above examples explain why there is or isn’t a tangent vector.

Worksheet 9: Functions of several variables

1. For f (x, y) = 6 − 3 x − 2 y (a) find the domain (include sketch) and range; (b) sketch some level curves; (c) sketch the graph.
2. For f (x, y) = √ 9 − x^2 − y^2 (a) find the domain (include sketch) and range; (b) sketch some level curves; (c) sketch the graph.
3. For f (x, y) = √x^2 + y^2 − 1 (a) find the domain (include sketch) and range; (b) sketch some level curves; (c) sketch the graph.
4. Find and sketch the domain of f (x, y) =

√x (^2) +y 3 x^2 +3x− 7.

5. Find and sketch the domain of f (x, y) = exp

( (^) x+y xy

6. Determine, as well as you are able, the level surfaces for f (x, y, z) = x^2 + y^2 − z^2.

Worksheet 10a: Limits and Continuity I

1. What does it mean for a function of a single variable f (x) to have a limit as x → a? What about to be continuous?
2. How would you try to define the notion of limits and continuity for a function f (x, y) of two variables? Three variables?
3. Can you give an example of a function f (x, y) whose limit does not exist at the origin?
4. Can you give an example of another function f (x, y) which is not continuous at the origin?
5. Should (x, y) → 0 mean the same thing as x → 0 and y → 0?
6. Should lim(x,y)→ 0 f (x, y) = limx→ 0 f (x, 0) = limy→ 0 f (0, y) if the limits exist?

Worksheet 11a: Partial Derivatives I

1. Recall what the definition of the derivative is for a function f (x) of one variable. What does it mean geometrically? Kinematically (in terms of motion)?
2. Can you think of a geometric analogue of derivative for a function f (x, y) of two variables?
3. Can you think of a kinematic analogue of derivative for a function f (x, y) of two variables?
4. How might you try to define derivatives for functions of two (or more) variables?

Worksheet 11b: Partial Derivatives II

1. Let f (x, y) = 3x + x^2 y^3 − 2 y^2. Find the first and second partial derivatives of f. Check fxy = fyx.
2. Let f (x, y) = x sin(xy). Find the first and second partial derivatives of f. Check fxy = fyx.
3. Find the first partial derivatives of f (x, y, z) = exy^ ln z.
4. For f (x, y) = 3x + x^2 y^3 − 2 y^2 , determine the tangent lines of the cross sections of z = f (x, y) for x = 0 and y = 0 at the origin. Determine the tangent plane to z = f (x, y) at the origin.

Worksheet 13: Extreme Values

1. Find the critical points of f (x, y) = 3x^4 + 3y^4 − 12 xy + 1 and classify them as hav- ing local minima or local maxima or being saddle points. Determine the absolute minimum and maximum of f (x, y) if they exist.
2. Find the distance from the point (1, 2 , 3) to the plane x + y + z = 10.
3. Find the maximum volume of a rectangular box such that the width, depth and height of the box sum to 12m.
4. Let R be the region in R^2 consisting of (x, y) such that 0 ≤ x, y ≤ 1. Can you exhibit a function f : R → R^2 such that f has no absolute maximum? Can you guess sufficient conditions on f to guarantee that is has or doesn’t have an absolute maximum?
5. Find the absolute maximum and minimum of f (x, y) = xy on the region x^2 + y^2 ≤ 1. What about the region x^2 + y^2 < 1.
6. Find the absolute maximum and minimum of f (x, y) = x^2 − 2 xy + y on the region of R^2 given by 0 ≤ x ≤ 3 and 0 ≤ y ≤ 2.

Worksheet 14: Double Integrals I

1. Compute ∫ ∫ R(x − 3 y^2 ) dA where R = [0, 3] × [1, 2].
2. Find the volume of the solid bounded by x^2 + 2y^2 + z = 16, x = 2, y = 2, and the 3 coordinate planes.

Worksheet 16: Double Integrals II

1. Give examples of regions that are (i) type I but not type II; (ii) type II but not type I; (iii) type I and type II; (iv) neither type I nor type II.
2. Find the volume of the tetrahedron bounded by x + 2y + z = 2, x = 2y, x = 0 and z = 0.
3. Compute ∫ ∫ R xy^2 dA, where R is region bounded by y = x − 1 and y^2 = 2x + 6.
4. Compute ∫ ∫ R sin(y^2 ) dA, where R is region enclosed by the triangle with vertices (0, 0), (0, 1) and (1, 1).

Worksheet 17: Double Integrals in Polar Coordinates

1. Find the volume of the region bounded by z = x^2 + y^2 and z^2 = x^2 + y^2.
2. Use a double integral to compute the area in R^2 of one loop of the curve r = cos θ.
3. Use a double integral to compute the area in R^2 enclosed by r = θπ , 0 ≤ θ ≤ 3 π and the line segment from (− 1 , 0) to (− 3 , 0).
4. (i) Compute volume under e−x^2 −y^2 over all of R^2 by using polar coordinates. (ii) Use (i) to compute ∫^ −∞∞ e−x^2 dx.