What is Calculus iii: Definition and 61 Discussions

Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space

R

3

.

{\displaystyle \mathbb {R} ^{3}.}
The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of
electromagnetic fields, gravitational fields, and fluid flow.
Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra which uses exterior products does (see § Generalizations below for more).

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1. What type of mathematics did you find the most difficult?

Mathematics tends to be more progressive than any other field. But I've heard some people say that some math classes that they took were more difficult than the most advanced math classes that they took. For instance, I've heard people say that Calculus II is more difficult than Calculus III...
2. Need help with a calculus III proof

I've been working on this one for a little bit, and I know I really just need to use the chain rule to solve it, but I can't seem to figure out how to set it up properly. Probably a dumb question, but I could really use some help on this!
3. Calculus III (help sketching graph)

Homework Statement Sketch the curve with the given vector equation. Indicate with an arrow the direction in with 't' increases r(t)=<t, 2-t, 2t>Homework Equations parametric equation (can't type the equation, too confusing to use the template) The Attempt at a Solution So far, I have <1, -1...
4. How to prove Convergence of this Series

Homework Statement Use any appropriate test to determine the convergence or divergence of the following series: \sum_{i=0}^{\infty} \frac{2^{i} + 3^{i}}{4^{i}+5^{i}} Homework EquationsThe Attempt at a Solution I've run it through mathematica and it told me it's convergent. However, I...

Dear Physics Forum personnel, I am a college sophomore in US with major in mathematics and an aspiring algebraist. I wrote this email because I am having a great problem with my current Calculus III (vector calculus) course that I am taking for this Summer Semester. All of my fellow...
6. Is f continuous at (0,0) and how to show it with curves?

Homework Statement https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-xap1/v/t1.0-9/10923273_407123639463753_2874228726948727052_n.jpg?oh=27c882da16071e65bbb420147333ec38&oe=558413E4&__gda__=1434978872_d03c8531060688181560956b68c96650 Is f continuous at (0,0)? What is the "maximum" region D...
7. Surface Integrals and Gauss's Law

When I learned Integrals in Calc III, the formula looked like this ∫∫ F(r(s,t))⋅(rs x rt)*dA but in physics for Gauss's law it is ∫∫E⋅nhat dA How are these the same basic formula? I know that nhat is a unit vector, so it is n/|n|, but in the actual equation, it is a dot between the cross...
8. Triple Integral of a cone bounded by a plane.

Homework Statement find the volume using spherical coordinates of the region bounded above by z=9 and below by z=sqrt(x^2+y^2) in the first octant. Homework EquationsThe Attempt at a Solution I found this volume using cartesian and cylindrical coordinates, so I know the answer I am looking...
9. Triple Integral, Volume of an Egg

Homework Statement I need to find the volume of an egg with a shape described by: z = 1/2(x2 + y2) and z = 6 - x2 - y2 I am also given that the egg is 6cm in length.Homework Equations I roughly graphed the two surfaces. The first being paraboloid that opens up from the origin, and the second...
10. Find a Rigorous Calculus of Multiple Variables Book for Advanced Learners

Hi! I am looking for a very rigorous book on some of the topics covered in Calculus of Multiple Variables. My University uses the last part of Adams "Calculus: a complete course" and I found the presentation therein more fit for people needing to know enough to perform the calculations than for...
11. Can you recommend me a good textbook for Calculus III?

I want a good textbook for Calculus III (Multivariable Calculus) that has lots of practice problems and less theoretical problems. The textbook must have lots of practice problems for multiple integrals, finding areas, volumes using multiple integrals, finding jacobians, gradients, partial...
12. Calculus III Problem: find smallest angle

Homework Statement Find the smallest angle between 3x-2y-4z = 5 and {x = 1 + 2t, y = 3 + 4t, z = 5 + 6t}. Homework Equations 3x-2y-4z=5 is a plane, the other 3 equations constitute a line. The Attempt at a Solution My understanding is that the angle between a plane and a line can...
13. MHB Calculus III is SO MUCH EASIER than Calculus II

I'm in Calculus III now, let me just tell you all - it is SO MUCH EASIER than Calculus II. I went through hell in Calculus II. Calculus III is so much easier that it's kind of ridiculous. What's Calculus 4 like?
14. Calculus III - Conceptual Help

I'm taking a Calc III Course and I want to know if anyone out there can help me with a few conceptual ideas. I know how to do the math but I am missing the conceptual idea of it. Why should/do we we parametrize? Why should/do we convert to Polar or Cylindrical? These basic ideas should be...
15. Calculus III; standard form of & angle to eliminate the cross produc

Hi; I've been trying to solve the problem myself but i really don't what could be wrong; The problem says : Make the change of variables x=ucos−vsin y=usin+vcos where the angle 0<(phi)<2 is chosen in order to eliminate the cross product term in x^2+xy+y^2=6 Then find the standard form of...
16. Vector Projectile Problem (Calculus III)

Homework Statement A golfer stands 390 ft (130 yards) horizontally from the hole and 40 feet below the hole. Assuming the ball is hit with an initial speed of 150 ft/s, at what angle should it be hit to land in the hole? Assume the path of the ball lies in a plane Homework Equations Force of...
17. MHB Is my Calculus III teacher really prepared for this class?

Well My calculus 3 teacher is well known for saying, "I Was not prepeared for this.." Evry single lessons.. When he show some proof he see that he makes misstake and use that as excuse.. And Also sometimes in exemple.. Then he says "You can read this exemple in book.." And I Dont even got same...
18. Calculus III Review for University of Arkansas Math Majors

I'm sure this question has been asked, but I'm unable to find where, but I was recently accepted to the University of Arkansas, majoring in Math and minoring in physics. I've taken Cal I-III already, but my course work is very sporadic, as I've changed majors once, and essentially taken two...
19. Calculus III: Find a line perdendicular to XY-plane?

Homework Statement [/b] "Find an equation for the line through the point P = (1, 0, −3) and perpendicular to the xy-plane," obviously this includes vector <0, 0, 1> I am in Calc III and need help understanding how to do this TYPE of problem. Please include step-by-step instructions and...
20. Calculus III - Multivariate Continuity

Homework Statement Let \begin{equation*} f(x,y) = \begin{cases} \dfrac{x^3 - y^3}{x^2 + y^2}, \hspace{1.1em} (x, y) \neq (0,0) \\ 0, \hspace{4em} (x,y) = (0,0) \end{cases} \end{equation*} Is f continuous at the point (0,0)? Are f_x og f_y continuous at the point (0,0)? Homework Equations...
21. Calculus III - Multivariate limit problem

Homework Statement Show that for all points (\bf{a,b}) from \mathbb{R}^n \times \mathbb{R}^n this applies: \displaystyle \lim _{(\bf{x,y}) \to ({\bf a,b})} \bf{||x-y|| = ||a - b||} Homework Equations Not sure. The Attempt at a Solution I thought about defining a and b as centers of...
22. Calculus III Tips, Help, or Advice

Hi, folks. I'm currently in Calc. III, and having a very hard time. Right now, each one of my home-work assignments seem to take hours. I'll spend 30 min. on one problem, get it wrong, do it again, and again, and again—until I finally get it right. It's taking huge amounts of time, and when I...
23. A contradiction? - Calculus III Question

yz = ln(x+z) So I'm trying to find the tangent plane to the surface at a particular point (x0,y0,z0). Here's the general formula: Fx(x0,y0,z0)(x-x0) + Fy(x0,y0,z0)(y-y0) + Fz(x0,y0,z0)(z-z0) = 0, where Fx, Fy, and Fz are the partial derivatives of the below F(x,y,z): 1. F(x,y,z) = ln(x+z)...
24. Calculating Volume of Intersecting Spheres | Calculus III Problem"

Homework Statement Find the volume of the solid that lies in between both of the spheres: x2+y2+z2+4x+2y+4z+5=0 and x2+y2+z2=4Homework Equations This is the first chapter of the calculus III material so no double or triple integrals are needed to solve this problem.The Attempt at a Solution I...
25. Can I Pass Calculus III Without Taking Calculus II First?

Hey guys, I'm facing a tough decision and I would like to ask you for some guidance and tips regarding my situation. I want to apply for econ grad school this fall and I'm missing some math essentials. The program I followed did not put so much emphasis on the mathematical background...
26. How Should I Prepare for Calculus III and Modern Algebra?

I'm taking both in the fall, what should I review over the summer to prep up for this?
27. Finding the Tangent Plane to a Surface at a Given Point

Homework Statement Suppose you need to know an equation of the tangent plane to a surface S at the point P(2,1,3). You don't have an equation for S but you know that the curves: r_1(t) = <2 + 3t, 1 - t^2, 3 - 4t + t^2> r_2(u) = <1 + u^2, 2u^3 - 1, 2u + 1> both lie on S. Find an...
28. Online summer classes, specifically Calculus III

Hey everyone, I've asked something similar, but the time to sign up for summer classes is nearing. Reason I want to take Calc III this summer- so I can take Modern Physics in the fall. I will be working 40 hours a week at a summer camp, so I'm not sure how that will pan out with an...
29. Any advice on taking Calculus III when Calculus II was hard?

I barely passed Calculus II, I got 50 in the whole course after finals and I had 56 before finals. The class avg was around a high D but I can't remember exactly now but that's besides the point. Calculus II at my college was integral calculus with some application of integrals(volume, surface...
30. Taking Calculus III and Real Analysis During Same Semester

Is it wise to take Calculus III and Intro to Real Analysis during the same semester? Or should I complete Calculus III and take Intro to Real Analysis afterwards? I ask because I do not want to stretch myself too thin, because I work over forty hours per week and have a family. If it makes...
31. Studying Any good Calculus III books that comes with explanations?

I am using Stewart's textbook right now and it is horrible. I really wonder how my other classmates get their work done, I feel like I have more trouble than them. Weird thing is, most of them don't even pay attention in class. Anyways, are there any books that give Calc III exercises and give...
32. Calculus III Stokes' Theorem

Hi everyone, I have to prove this problem but I have no idea how to approach this problem. I tried something but it seems not working... Suppose F is a vector field in R3 whose components have continuous partial derivatives. (So F satisfies the hypotheses of Stoke's Theorem.) (a)...
33. Calculus III or Linear Algebra? Which one should I take first?

Firstly, as you may see, I'm a newbie to this forum so I apologize if I posted this question in the wrong place or violated any rules within this post. :) I have no idea as to which to take first. The course description for Linear Algebra says that "Calculus III is recommended but no...
34. Calculus III or linear Algebra? Which one should I take first?

Firstly, as you may see, I'm a newbie to this forum so I apologize if I posted this question in the wrong place or violated any rules within this post. :) I have no idea as to which to take first. The course description for Linear Algebra says that "Calculus III is recommended but no required"...
35. Will I be allowed to skip Calculus II if I do this? I passed Calculus III

This is my situation, last week I took the AP Calculus BC exam, I feel like I did really poorly and even if they are going to curve it, I think I am going to end up with a 3 or 4. Now in this summer, I going to take a Calculus III at a college who has granted me exemption from prerequisites...
36. Fall 2010 - Calculus III or Linear Algebra

I am a computer science major with a minor in mathematical sciences. I just finished up Calculus I this semester (with a 97%, yay!) and scheduled to take Calculus II this summer. The list of available fall classes was just released and I'd like to take either Calculus III or Linear Algbera. I...
37. What is a good book for Multi-variable Calculus III?

OKay, so I self-studied Calculus II when I was in High school using a prep book and I did very well. So I am looking into self-studying Calculus III, what good books are there? Do not suggest textbooks because I didn't use textbooks when I self-studied Calculus II.
38. What is the derivative of a vector function in Calculus III?

Given that X is a vector in R^n, what is the derivative of: \texts{f(X)=|X|^{2}X} I basically combined the product formula and dot product rule after breaking down |X|^2, which yielded my answer of: \texts{f'(X)=|X|^{2}v+2(v\bullet X)X} Where v the direction. Is this correct/ close?
39. Calculus III book with focus on problems.

We're currently using Stewart's in my class, and I'm not at all happy with it. I actually liked Stewart's for single variable, but the multivariable volume feels like it rushes over all the concepts and provides very little insight, and the problems are WAY too easy. They are useless for...
40. Calculus III: Derivatives and composition

I need prove or disprove the following statement: If f: R^2->R^2 is a differentiable function whose derivative at (0,0) is not invertible, then there does not exist a differentiable function g: R^2->R^2 such that: (g o f)(X)=X I've been trying to find counter examples like crazy...
41. Proof of Calculus III: Proving Limit as (x,y) -> (0,0) = 0

Hi! I was wondering if someone could give me a couple hints on how to tackle the following proof! Let f(x,y)= [ (lxl ^a)(lyl^b) ]/ [(lxl^c) + lyl^d] where a,b,c,d are positive numbers. prove that if (a/c) + (b/d) > 1 then limit as (x,y) -> (0,0) of f(x,y) exists and equals zero. thanks!
42. Calculus III: Open sets proof help

I need to prove that the following is an open subset of R^2: \left\{(x,y)\inR^{2}|\sqrt{x^2+y^2}<1} I think the substition r=min{sqrt[x^2+y^2],1-sqrt[x^2+y^2]} works, but I'm stuck on how to take it from that to showing that the distance between X0 and X1 is less that r, and more...
43. Converging to 5: Solving a Tricky Calculus III Series Problem

I've been working on this for two hours and have had zero luck: Given: sum{k=1 to k=oo} [((-1)^(k+1))/k] Rearrange the terms so the series converges to 5 [lol, I haven't a clue how].
44. Basic Calculus III- Arc Length Parameter and Length- Getting a negative length

Homework Statement Find the Arc Length Parameter along the curve from the point where t = 0 by evaluating the integral: s = ∫ |v(τ)| dτ from 0 to t Then find the length of the indicated portion of the curve. Homework Equations The vector I am using for this: r(t) = (etcos t)i + (etsin...
45. Should one take Calculus III, Linear Algebra, DiffyQ ,E&M all in semester?

Am I setting myself up for disaster? Here is my current Fall 09 schedule: Calculus III (Honors) differential equations (Honors) E&M (Would do honors if they offered the section this Fall) Linear Algebra (Honors) Mind you, this is at a community college (so think of honors as somewhat...
46. Linear Algebra or Calculus III

I need some advice... I have the option to take Linear Algebra from an excellent professor, but the course isn't required for my physics major and would only count as an elective for my math minor. Alternatively, I could take Calculus III from a mediocre (at best) lecturer. This course is...
47. Maximizing Success in Self-Studying Calculus III: Tips and Warnings

i'm planning on taking this course at a university in fall along with a large load of other courses, but i want to get a headstart. So i bought the book, would you recommend reading the text, reading a friend's notes and just doing the homework by myself? will this give me an advantage when i...
48. Finding the Area of an Ellipse in Calculus III

Homework Statement Find the area of the ellipse (x^2)/4 + (y^2)/9 = 1 Homework Equations (1/2) * Integral of xdy - ydx = area of a Region The Attempt at a Solution The problem is weird to me because its complicated to attempt it in rectangular or polar coordinates)
49. Calculus III Write the equation of the plane

Homework Statement Write the equation of the plane that contains the following two parrallel lines R(t) = <1,0,3> + t<1,,4,-2> and E(t) = <2,3,0> + t<1,4,-2> Homework Equations The Attempt at a Solution I tried parameterizing of the equations that didnt seem to work. Please...
50. Calculus III : Getting MacLaurin Series From Functions

Homework Statement 5) For each of the series below, write the series in summation notation and give the first five terms of the series. Also give the radius of convergence of the series. a) Use the series for \frac{1}{1 - x} to find the Maclaurin series of f(x) = \frac{1}{(1-2x)^3}...