# What is Calculus 3: Definition and 82 Discussions

Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread uses in science, engineering, and economics.In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus (plural calculi) is a Latin word, meaning originally "small pebble" (this meaning is kept in medicine – see Calculus (medicine)). Because such pebbles were used for counting (or measuring) a distance travelled by transportation devices in use in ancient Rome, the meaning of the word has evolved and today usually means a method of computation. It is therefore used for naming specific methods of calculation and related theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus.

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1. ### Surface Integral of a sphere

Solving the integral is the easiest part. Using spherical coordinates: $$\oint_{s} \frac{1}{|\vec{r}-\vec{r'}|}da' = \int_{0}^{\pi}\int_{0}^{2\pi} \frac{1}{|\vec{r}-\vec{r'}|}r_{0}^2 \hat r \sin{\theta}d\theta d\phi$$ then: I = \dfrac{1}{|\vec{r}-\vec{r'}|}r_{0}^2(1+1)(2\pi)\hat...
2. ### Confused about polar integrals and setting up bounds

So I want to subtract the two surfaces, right? I really don't know where to start... I am guessing this would be some sort of triple integral, however I am very confused with the bounds. Any help would be greatly appreciated! Thanks!!

Pending
4. ### Calculus Looking for a rigorous multivariable calculus book

Hello everyone. I'm about to take Calc 3 next semester and am looking for a rigorous book to work with on multivariable calculus. I've gone through Spivak's "Calculus" from cover to cover and am hoping to find something with the same degree of rigor, if possible, and preferably with a solution...
5. ### MHB Calculus 3 Help: Iterated, Double, Triple Integrals & More

Kindly help me with: Iterated integrals Q1, 2,3 Double integrals in polar coordinates Q1, 2,3 Triple integrals Q1, 2,3 Triple integrals in cylindrical coordinates Q1, 2,3 Triple integrals in spherical coordinates Q1, 2,3 Change of variables Q7,8,9 Green's theorem Q1,2 Surface integrals...
6. ### I Multivariable limits - path problem

Hey, so I have the following problem: I'm trying to prove that the limit doesn't exist (although I'm not sure if it does or not) so: along y=mx -> x=y/m: , which is 0 for all k≠0. along y^n it's the same and I'm not sure what I should do next. Could I set x = sin(y)? If I can, then the limit...
7. ### Studying Ordinary Differential Equations and Calc III

Hello, I need help deciding on whether to take ODE (MAP2302) and Calc III during the summer. Would it be wise to take ODE along with Calc III in the same semester? Some people have told me to take Calc III first because there are a few things in ODE that are taught in Calc III, but others have...
8. ### What is the implicit differentiation of the van der Waals equation?

Summary:: van der waals I have a pretty good understanding of implicit differentiation. However I'm stuck on a homework problem and could use some help. [P + (an^2)/V^2][V - nb] = nRT a,n,b,R are constants My professor wants me to take the implicit differentiation of P wrt...
9. ### Finding an area of a triangle formed by three points

Homework Statement P(3, 0, 3), Q(−2, 1, 5), R(6, 2, 7) (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. (b) Find the area of the triangle PQR Homework Equations A = \frac{1}{2}|\vec{AB}\times\vec{AC}| Source...
10. ### Problem solving this volume using Jacobi's Determinant

Homework Statement Find the value of the solid's volume given by the ecuation 3x+4y+2z=10 as ceiling,and the cilindric surfaces 2x^2=y x^2=3*y 4y^2=x y^2=3x and the xy plane as floor.The Attempt at a Solution I know that we have to give the ecuation this form: ∫∫z(x,y)dxdy= Volume So, in fact...
11. ### Converting Cartesian to Polar (Double Integral)

Homework Statement Integrate from 0 to 1 (outside) and y to sqrt(2-y^2) for the function 8(x+y) dx dy. I am having difficulty finding the bounds for theta and r. Homework Equations I understand that somewhere here, I should be changing to x = r cost y = r sin t I understand that I can solve...
12. ### Equations of two concentric paraboloids

Homework Statement I need to obtain the equation of 2 paraboloids separated by a distance L. Homework Equations I think that the equations should be: z_1=x^2+y^2 z_2=x^2+y^2-L The Attempt at a Solution The problem is that when I plot the region between two inequations, x^2+y^2>=z and...
13. ### Calculus *BEST* Calculus 3 Textbook for self study

Hello, currently I am a high school senior who will be going to college in the fall and since my school ends in may and college starts in mid-August. I am planning on self-studying calculus 3, so I can test out of it and go straight into partial differential equations. The textbooks that the...
14. ### I Overcoming Struggles in Calculus 3

I am currently nearing the end of my Calculus 3 course and have been struggling all semester. First there is some background information. I passed Calc 1 and 2 with a B and C respectively. Over the summer I worked on my skills and felt prepared. Unfortunately my section was chosen for IBL...
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### Calculus 3 help -- Is the gradient of a plane the normal to the plane?

Homework Statement Is the gradient of a plane, the normal to the plane? If so, why? Homework Equations No idea, just a question that popped up in my head eon of plane: n(x-x1)+n(y-y1)+n(z-z1) The Attempt at a Solution I found the partial derivative of each, and got the normal.[/B]
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### Calculus 3 help -- The equation of a plane and finding a point on that plane

Homework Statement Why is that we can set two variables zero in an equation of a plane to find a point on that plane? What is the proof for this? Homework EquationsThe Attempt at a Solution
17. ### MHB Why Is Problem #4 in Calculus 3 Series So Challenging?

i have attached the problem set. I have done the first three problems but number 4 is very difficult. Can someone help me out? Thanks [Editor's note: The PDF below contains the complete problem set from which #4 is as shown above.]
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### Calc 3: Equation of a line normal to the contour

Homework Statement Given g(x,y)=x2 - xy + 2y2, find the equation of the line normal to the contour that passes through the point (1,2). Homework Equations Not 100% positive, but the equation to a plane tangent to a function of 3 variables g(x,y,z) is (partial of x)(x - x0) + (partial of y)(y -...
19. ### Courses Calculus 3 -- looking for ways to help me understand

So I am in calculus 3 this year and have passed both calc 1 and 2 with a B and C+ respectively. I could have gotten a better grade but was lazy. I was lazy by using calculators and not actually learning the arithmetic and algebra. Now one serious issue I have is Trig. I can never remember trig...
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### Faraday's law with Calc 3 integration help

Homework Statement Suppose an infinitely long wire carrying current ##I=sin_0(\omega t)## is a distance ##a## away from a equilateral triangular circuit with resistance ##R## in the same plane as shown in the figure. Each side of the circuit is length ##b##. I need to find the induced voltage...
21. ### Need help with this vector problem -- Thank you

Homework Statement Let L1 be the tangent line to r(t) at the point t = a and let L2 be the tangent line where t = b. Find the equation of the lines L1. Find the equation of the lines L1 and L2 and find the points of intersection. r(t) = <f(t), g(t), h(t)> *bolded letters are vectors Homework...
22. ### Osculating Circle Homework: Solving for Curvature Limit

Homework Statement I know I would use the curvature equation |f''| / [1-(f')^2]^3/2 and then take the limit of that to -1. I just don't understand why I have to take the limit of the curvature and when I take the limit of the curvature I get |-1| / (13)^3/2 when the answer should be 2.
23. ### Partial Derivatives: Solve Homework Quickly

Homework Statement So I know I have to take the derivative with respect to x, then respect to y, then respect to z, but I am not getting the right answer. I know that the answer is 0 and my professor did it with very few steps that I do not understand. Can someone please guide me through it?
24. ### Finding the curvature of a space curve

Homework Statement Find the curvature of the car's path, K(t) Car's Path: r(t) = \Big< 40cos( \frac {2 \pi}{16}t ) , 40sin( \frac {2 \pi}{16}t ), \frac{20}{16}t \Big> Homework Equations K(t) = \frac { |r'(t)\:X \:r''(t)|}{|r'(t)|^3 } The Attempt at a Solution This is part of a massive 6...
25. ### Calculus 3 change of variables

Homework Statement Use the change of variables to evaluate the integral (x +y ) sin(x -y )dA, where R is the region enclosed by y = x, y = x - 2, y = -x and y = -x + 1. (Hint: use u = x + y and v = x - y Homework EquationsThe Attempt at a Solution Not sure how to start it
26. ### Help with Multi-variable Limits

Homework Statement Evaluate or show that the limit does not exist: \lim_{(x,y) \to (0,0)}\frac{ 2x^{4} + 5y^{3} }{8x^{2}-9y^{3}} \lim_{(x,y) \to (0,-2)}\frac{ xy+2x }{3x^{2}+(y+2)^{2}} Homework EquationsThe Attempt at a Solution So the first one is indeterminate and cannot be factored...
27. ### Calculating the Mass of a Half Ball with Given Density δ(x, y, z)

Homework Statement If the density of the half-ball x ^2 + y ^2 + z ^2 ≤ 4 ; z ≥ 0 is given by δ(x, y, z) = ( x^ 2 + y^ 2 + z ^2)^(1/2) find its mass. Homework Equations ∫∫F⋅ds ∫∫FoΦ(u,v)||Tu,Tv||dudv ∫∫FoΦ(u,v)⋅(Tu,Tv)dudvThe Attempt at a Solution For the last problem I was asked to find the...
28. ### Calc III Double Integral Question

This is the problem I'm trying to solve. The directions require me to rewrite as a single integral and evaluate. But I'm having trouble setting the bounds for a complete compounded integral. The graph of the region would look something like this... Where the shaded area is the region. I...
29. ### Determine the Set of Points at Which the Function is Continu

Homework Statement "Determine the set of points at which the function is continuous" ƒ(x,y,z) = √y/(√(x2-y2-z2)) Homework Equations ƒ(x,y,z) = √y/(√(x2-y2-z2)) The Attempt at a Solution If y is zero this function would be discontinuous or if the denominator became zero. As much detail as you...
30. ### Contour map of the function showing several level curves

Homework Statement ƒ(x,y) = ln(x2+4y2) Homework Equations I'm not really sure but I solved for y The Attempt at a Solution
31. ### Find and Sketch the Domain of a Function

Homework Statement ƒ(x,y) = √y + √[25-x2-y2] Homework Equations Quadratic The Attempt at a Solution The expression for f(x,y) is defined as long as x2 + y2 ≥ 25 and y ≥ 0 because of the square roots we can't have negative numbers. {(x,y) ∈ ℝ2 : y ≥ 0 and x2 + y2 ≥ 25} I'm not sure how I'd...
32. ### Show the gravitional field is conservative

In my calculus textbook (section on vector calc) it is showing that the gravitational field is conservative. I followed fine except for the first part, defining the scalar function f. Showing the field is conservative went something like this: f(x,y,z) = MM'G/\sqrt{x^2+y^2+z^2}...
33. ### Volume bounded by two surfaces, what am I missing?

Homework Statement Find the volume of the solid bounded by z=x^2+y^2 and z=8-x^2-y^2 Homework Equations use double integral dydx the textbook divided the volume into 4 parts, The Attempt at a Solution [/B] f(x)= 8-x^2-y^2-(x^2+y^2)= 4-x^2-y^2 i use wolfram and got 8 pi, the correct...
34. ### How Can dv/dx Be Determined to Solve for dv/dt?

Homework Statement Homework Equations Chain rule, partial derivation The Attempt at a Solution dv/dt=dv/dx*dx/dt+dv/dy*dy/dt dx/dt=-4t -> evaluate at (1,1) =-4 dv/dt=-4dv/dx+4(-2) dv/dt=-4dv/dx-8 How can I find the missing dv/dx in order to get a value for dv/dt? Thanks!
35. ### Multivariable continuity using limits

Homework Statement Homework Equations lim(x,y)->(a,b)f(x,y) continuous at (a,b) if lim(x,y)->(a,b)f(x,y)=f(a,b) Squeeze theorem if lim a=lim c and lim a<= lim b <= lim c then lim b= lim c The Attempt at a Solution I proved that all the limits exist but somewhat the functions aren't all...
36. ### Calculus Theoretical Multivariable Calculus books

Dear Physics Forum advisers, Could you recommend books that treat the multivariable calculus from a theoretical aspect (and applications too, if possible)? I have been reading Rudin's PMA and Apostol's Mathematical Analysis, but their treatment of vector calculus is very confusing and not...
37. ### Precalc topics that are essential in calculus 3

Hello, I will be taking calc3 a semester from now. I will take linear algebra this upcoming semester, so i have a bit of time to prepare for calc. what are Precal topics that are essential in calculus 3? reading the class description, it seems more geometric than calc 1 and 2. Also, what are...
38. ### How Do You Find a Vector with a Specific Magnitude in the Opposite Direction?

Homework Statement Find a vector in the direction opposite to <-4,1,2>, that has a magnitude of 3. Homework Equations I think that I did the first part of the problem correctly: <-4,1,2> magnitude= sqrt[ (-4)^2+1^2+2^2 ] = sqrt(16+1+4) = sqrt(21)...
39. ### Are hyperbolic functions used in Calculus 3?

More than just a few problems that happen to pop up in the textbook, I mean.
40. ### Partial Differentiation -- y deleted or ignored?

Homework Statement Given: z = f(x,y) = x^2-y^2 To take the partial derivative of f with respect to x hold y constant then take the derivative of x. ∂f/∂x = 2x What I don't understand is why such would equal 2x, when the y is still there it just isn't variable and is ignored. Wouldn't it be...
41. ### Calculus 3 vs Differential Equations

My school requires me to take a calc 3 course and differential equations for my major. I am scheduled right now to take differential equations in the summer, and calc 3 in the fall of this year. My school recommends they be concurrent classes, but I'm just too strapped down by my already almost...
42. ### Find the average height of a pyramid

Homework Statement Homework Equations V = (1/3) * A * H [Volume of Pyramid] The Attempt at a Solution The first thing I did was to calculate the height of pyramid from the volume formula. I got a perpendicular height of 15. I'm not sure where to go from there. I'm under the impression...
43. ### Use vectors and the dot product to prove the midpoint

Homework Statement [/B] Use vectors and the dot product to prove that the midpoint of the hypotenuse of a right triangle is equidistant to all three vertices. Homework Equations [/B] I know the dot product is A⋅B = |A||B|cosΘ ... or ... A1B1 + A2B2 + A3B3 ... + AnBn I know the...
44. ### Programs Calculus 3 or LA/DQ to help decide a major

Hello everyone, I've read advice on here about waiting to decide on a major until you have taken most of the math and physics courses. I am here to ask which math class I should take in order to help me decide which major would be best for me? I am debating still between math/physics/ee and...
45. ### Calculus 3 problem: lines and planes in space

Let u=<5,-2,3> and v=<-2,1,4>. Find the value of c which will force the vector w=<2c,3,c-1> to lie in the plane of u and v. I did the cross product of u and v, then i crossed u and w, then I equal the product of u and v with what I got for w. But for some reason when I try doing the triple...
46. ### Calculus 3 -- should I take linear algebra before or can I take it after?

I will start calculus 2 in the fall. I am comfortable with the integration techniques. I did not quite get Taylor series an d the like during self study. I can learn this from the professor swing that I have somewhat completed the cal 2 portion and the semester is 2 months away. now my...
47. ### Calculus 3: Work Homework: Electric Dipole Field

Homework Statement An electric dipole with dipole moment p = 4 × 10−5 C-m sets up an electric field (in Newtons per coulomb) F(x,y,z) = kp/r^5 <3xz, 3yz, 2z^{2} - x^{2} - y^{2}> where r = (x^{2} + y^{2} + z^{2})^{1/2} with distance in meters and k = 8.99 × 109 N-m^{2}/C^{2}...
48. ### Is f(xy)=e^(2y)sin(pix) a Solution to the Differential Equation Given?

let f(xy)=e^(2y)sin(pix), is f a solution to (fxy)^2 -fxx(fyy)=4pi^2 e^(4x) So I found fxx and fyy and fxy, which are -pi(e^(2y))sin(pix), 4sin(pix)e^(2y), 2pi(e^(2y))cos(pix) respectively, When i reduced everything i got e^4y(sin^2(pix))+pie^4ycos^2(pix)=pie^(4x). I am assuming it is...
49. ### Good opencourseware resource for Calculus 3?

Next semester, I need to take Calc 3 to finish my AA. However, our college will not be offering it at that time, though they know of an online course I can take. Since I know an online course is unlikely to provide the support needed to really get the material down, I am looking for a decent...
50. ### What should I study after Calculus 3?

Hey guys, So I am a physics and math major so far I am took Calculus I,II,III linear algebra and statistics I did really good in them almost acing all of them and aced vector calculus. I am taking complex analysis and introduction to probability in fall. I have holiday in the winter...