# What is Multivariable: Definition and 536 Discussions

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather than just one.

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1. ### Operators On Multivariable Wave Functions

I know the way to solve the first part is to find <ψ|Αψ> and compare it with <ψΑ|ψ>. This comparison can be done through an integral representation where we take ψ* and act A on ψ to be the integrand, or act A on ψ* and multiply by ψ for the integrand. If the integrals are the same, then the...
2. ### I Differentiability of a Multivariable function

I’m having a little confusion about part b of this question as to why I am allowed to use the limit definition of a partial derivative. Here’s what I think: I know that y^3/(x^2+y^2) is undefined at the origin but it does approach 0 when it GETS CLOSE to the origin. So technically defining...
3. ### Multivariable calculus problem involving partial derivatives along a surface

I just wanted to know if my solution to part (b) is correct. Here's what I did: I took the partial derivative with respect to x and y, which gave me respectively. Then I computed the partial derivatives at (-3,4) which gave me 3/125 for partial derivative wrt x and -4/125 for partial derivative...
4. ### Calculus Does Apostol Calculus Volume 2 cover sufficient multivariate calculus?

Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient...
5. ### Calculus Multivariable calculus PDF books

Multivariable calculus is a branch of mathematics that extends the concepts of single-variable calculus to functions of multiple variables. In this subject, vectors and partial derivatives are introduced to represent and manipulate multi-dimensional data. The gradient of a function represents...
6. ### I Differential operator in multivariable fundamental theorem

I'm referring to this result: But I'm not sure what happens if I apply a linear differential operator to both sides (like a derivation ##D##) - more specifically I'm not sure at what point should each term be evaluated. Acting ##D## on both sides I'll get...
7. ### I Multivariable function optimization inconsistency

Mentor note: For LaTeX here at this site, don't use single \$ characters -- they don't work at all. See our LaTeX tutorial from the link at the lower left corner of the input text pane. I have a function dependent on 4 variables ##f(r_1,r_2,q_1,q)##. I'm looking to minimize this function in the...
8. ### Calculating total derivative of multivariable function

This isn't a homework problem exactly but my attempt to derive a result given in a textbook for myself. Below is my attempt at a solution, typed up elsewhere with nice formatting so didn't want to redo it all. Direct image link here. Would greatly appreciate if anyone has any pointers.
9. ### Critical points of a multivariable function (wave equation)

I would like to check my understanding of this problem. There are the following possibilities: a. Isolated points where the gradient is 0. b. The level curves of height 0 c. The level curves of height 1. d. The level curves of height -1. e. None of the above. I would choose a, c, d. Where...
10. ### Good approximation - multivariable calculus

I tried to use a Taylor series expanded at 3 and set to 3.01: https://www.wolframalpha.com/input/?i=27+++9+(-3+++x)^2+++(-3+++x)^3+++3+y^2+++y^3+++(-3+++x)+(27+++y^2)=3.01 I got the vector ## (\Delta x, \Delta y)= (0.37887, -0.54038)## It does give a desired result but it is marked as wrong...
11. ### Multivariable temperature variation while swimming in a hot spring

I have computed ##T_x## and ##T_y## and evaluated it at the point (20, 20). ## \frac {-450*x}{x^2 + y^2 + 1)^(\frac {3} {2}} - \frac {420*(x + 10)} {(x + 10)^2 + (y - 20)^2 + 1)^(\frac {3} {2}}, \frac {-450*y}{x^2 + y^2 + 1)^(\frac {3} {2}} - \frac {420*(y - 20)} {(x + 10)^2 + (y - 20)^2 +...
12. ### Domain of multivariable function - definition

E.g. all real numbers could be a domain but not necessarily, etc. Am I right?
13. ### Analysis Multivariable analysis textbooks

Hi! the next semester i'll take a course on multivariable calculus but i'd like to study it now. I've "Principles of mathematical analysis" by W.Rudin but i heard that the sections about this are hard to read. If there is a textbook like "Understading analysis" by Stephen Abbott but in...
14. ### Understanding Multivariable Limits: Solving with Factoring Methods

I do not understand how they got the -x in the numerator to turn into a sqrt(x) when factoring to solve this multivariable function. Could some help me understand?
15. ### Multivariable Calculus proof for Optics

Part A) For part A I forgo breaking down the identity into it's component x, y, and z parts, and just take the r derivative treating r' as a constant vector. This seems to give the right answer, but to be entirely honest I'm not sure how I'd go about doing this component by component. I figure...
16. ### 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...
17. ### Help solving this Heat Equation please

I want to solve the heat equation below: I don't understand where the expression for ##2/R\cdot\int_0^R q\cdot sin(k_nr)\cdot r \, dr## came from. The r dependent function is calculated as ##sin(k_nr)/r## not ##sin(k_nr)\cdot r##. I don't even know if ##sin(k_nr)/r## are orthogonal for...
18. ### B Is multivariable calculus fun?

I've been studying calculus A and B on and off over the last ten years, and I'm starting to learn calculus again for fun as soon as I can get my hands on a textbook. I was wondering if multivariable calc is as fun as A and B have been so far.
19. ### Integrating Mass of a Hollow Sphere: Multivariable Calculus Explained

I know some multivariable calculus, I just want someone to walk me through the integration deriving the mass element dM and the integration of thin rings composing the hollow sphere. It would also be nice if you could show me doing it one way using the solid angle and one way without using the...
20. ### 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...
21. ### I Multivariable optimization problem

Hi all, (Please move to general or mechanical engineering sub-forum if more appropriate over there. I put this here as it is essentially a mathematics problem.) Broken into sections: - problem categorization (what type of problem I think I have), - the question, - specifics (description of the...
22. ### MHB Diff-in-Diff Regression: 6 Var, Interaction Terms & Estimate

I am doing a difference-in-difference analysis on a set of survey data for a health education program and I need to find statistical significance for the difference-in-difference estimate. I know that I find this using a regression. I need to use a regression in a mixed logistic model including...
23. ### Multivariable Limit Problem: Find Values of k That Make Limit Exist

(a) I thought perhaps a parameterization would be the place to begin given all the squared terms. x=rcos(u)sin(v) y=rsin(u)sin(v) z=rcos(v) That would yield: r^k(cos(v))^k*(e^(r^2*(sin(v))^2))/(r^(2k)) Canceling a r^k at each level: (cos(v))^k*(e^(r^2*(sin(v))^2))/(r^(k)) I'm not sure how...
24. ### Maximization of a Multivariable Function

The beginning is straight forward and I found f=x^2-2yz, which satisfies grad(f)=F. Then I calculated W= f(x,y,z)-f(0,1,1) since it's conservative. I get stuck when trying to find the max and mins. Given grad(f)=0 at extrema, we can see (0,0,0) is a point. On the boundary, I have to...
25. ### Multivariable Arc Length Problem: Weird Form with Parameterization

Problem: See Attachment. Parts (a) & (b) are clear, but my confusion arises in (c)-- I feel like there is a much simpler form. While technically my answer is correct, there must be something I'm missing. I parameterized the curve C=(t, e^2t, e^2t) and got c'(t)=(1,2e^2t,2e^2t), which should...
26. ### Calculus What book should I get for multivariable calculus after Stewart?

Hi. I just finished the single variable part of Stewart's calculus book which helped me to master AP calculus. Now I am planning to move on to non-rigorous multivariable calculus. However, I have found reading his book a bit painful since the book mainly focuses on problem-solving techniques...
27. ### Multivariable Calculus, Line Integral

The vector field F which is given by $$\mathbf{F} = \dfrac{(x, y)} {\sqrt {1-x^2-y^2}}$$ And the line integral $$\int_{C} F \cdot dr$$C is the path of $$\dfrac{\ (\cos (t), \sin (t))}{ 1+ e^t}$$ , and $$0 ≤ t < \infty$$ How do I calculate this? Anyone got a tip/hint? many thanks
28. ### I The domain of a multivariable function

hey there I'm struggling on finding the domain of the following function log (xy2)+x2y) I then do xy(y+x)>0 but then i don't know what to do with xy one attempt \begin{cases} y+x>0\\ x>0\\ y>0 \end{cases} union \begin{cases} y+x<0\\ x<0\\ y<0 \end{cases} but this doesn't lead to the...
29. ### Finding the limit of a multivariable function

If one approaches the origin from where ##x_2=0##, the terms ##x^2_1x_2+x^2_2x_3## in the denominator equal ##0##. Substituting ##|\textbf{x}|^2## for ##t## yields the expression ##\frac{e^t-1}{t}##, which has limit 1 as ##\textbf{x}\to\textbf{0}## and thus ##t\to0##. So the limit should be 1 if...
30. ### I Showing that a multivariable limit does not exist

I want to show that the limit of the following exists or does not exist: When going along the path x=0 the limit will tend to 0 thus if the limit exists it will be approaching the value 0 when going along the path y=0, we get an equation with divisibility by zero. Since this is not possible...
31. ### Developing a multivariable function

Hello! I am facing a difficulty into developing a multivariable function of a dependent variable "x". Let's assume that "x" is a function of 6 independent variables a,b,c,d,e,f,g. From experimental data i have developed 6 functions, each representing how "x" changes by each of the paremeters...
32. ### Multivariable Triple Integral - Calculus Physics/Math Problem

Hello everybody. If anyone could help me solve the calculus problem posted below, I would be greatful. Task: Evaluate the moment of inertia with respect to Oz axis of the homogeneous solid A Bounded by area - A: (x^2+y^2+z^2)^2<=zSo far I was able to expand A: [...] so that I receive...
33. ### Multivariable Chain Rule Question

For context, we have an equation f(x,y) = \frac{x}{y} and we had used the substitutions x = r \cos\theta and y = r \sin\theta . In the previous parts of the question, we have shown the following result: \frac{\partial f}{\partial x} = \cos\theta \Big(\frac{\partial f}{\partial r}\Big) -...
34. ### MHB McInerney Example 3.1.5: Multivariable Differentiation Q&A

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.1: The Derivative and Linear Approximation ... In Section 3.1 McInerney defines what is meant by...
35. ### I Multivariable Differentiation .... another example .... McInerney Example 3.1.5 .... ....

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.1: The Derivative and Linear Approximation ... In Section 3.1 McInerney defines what is meant...
36. ### I Multivariable Differentiation .... McInerney Definition 3.1.1 ....

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.1: The Derivative and Linear Approximation ... I am trying to fully understand Definition 3.1.1 and...
37. ### MHB Multivariable Differentiation .... McInerney Definition 3.1.1

I am reading Andrew McInerney's book: First Steps in Diofferential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.1: The Derivative and Linear Approximation ... I am trying to fully understand Definition 3.1.1...
38. ### I Is there such a thing as an antiderivative of a multivariable function?

Is there such a thing as an antiderivative of a multivariable function? I haven't put too much thought into this yet but I wanted to ask anyways. Sticking for now just to two variables, I was observing that double integrals are always definite integrals, whereas in the single-variable case, we...
39. ### A Multi-variable function depending on the Heaviside function

How can I calculate ∂/∂t(∫01 f(x,t,H(x-t)*a)dt), where a is a constant, H(x) is the Heaviside step function, and f is I know it must have something to do with distributions and the derivative of the Heaviside function which is ∂/∂t(H(t))=δ(x)... but I don't understand how can I work with the...
40. ### Limit of a multivariable function

Homework Statement If possible, calculate the following limit: \lim_{(x,y)\rightarrow (0,0)} {\frac{2x^2 + 3y^2}{5xy}} Homework Equations N/A The Attempt at a Solution [/B] I tried using both parametric and polar equations to find the limit, but neither worked. Setting either x or y...
41. ### Calculus Multivariable calculus without forms or manifolds

Hi there all, I'm currently taking a course in Multivariable Calculus at my University and would appreciate any recommendations for a textbook to supplement the lectures with. Thus far the relevant material we've covered in a Single Variable course at around the level of Spivak and some Linear...
42. ### B How do you create a + and π sign using multivariable (x,y,z)

I am taking a high school multivariable calculus class and we have an end-of-semester project where we trace out some letters etc., except that they all have to be connected, continuous and differentiable everywhere. My group's chosen to do Euler's formula, but so far we are having problems...
43. ### Multivariable calculus problem

Homework Statement Find the points on the surface xy^2z^3=2 that are closest to the origin Homework EquationsThe Attempt at a Solution x,y,z=/= 0, as when x,y,z = 0 it is untrue. Right?? Otherwise, I am very unsure as to how to approach this problem. Should I be taking partial derivatives to...
44. ### Multivariable Calculus, plane sketching

How do I know where to put the axes for the equation 4x^2 - 9y^2 = z when graphing in 3d?
45. ### I The Chain Rule for Multivariable Vector-Valued Functions ....

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.9: The Chain Rule ... ... I need help in order to fully understand Theorem 12.7, Section 12.9 ... Theorem 12.7...
46. ### MHB The Chain Rule for Multivariable Vector-Valued Functions .... ....

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.9: The Chain Rule ... ...I need help in order to fully understand Theorem 12.7, Section 12.9 ...Theorem 12.7...
47. ### MHB Differentiability of Multivariable Vector-Valued Functions .... ....

In Theodore Shifrin's book: Multivariable Mathematics, he defines the derivative of a multivariable vector-valued function as follows: Lafontaine in his book: An Introduction to Differential Manifolds, defines the derivative of a multivariable vector-valued function slightly differently as...
48. ### Calculus Resources for learning multivariable calculus

I have recently started studying multivariable calculus and I cannot quite visualise the concepts.Problem solving is not a problem but I want a true understanding of the concepts.Which book or online resources are great at developing visualisation in this course?
49. ### I Chain Rule of Multivariable Calculus

I am confused when I should use the ∂ notation and the d notation. For example, on http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx, in Case 1, the author wrote dz/dt while in Case 2, the author wrote ∂z/∂t. Could anyone please explain to me when I should use the ∂ notation and the...
50. ### Boas 4.12.18, 2nd Derivatives of Imp. Multivariable Integral

Homework Statement Show that u(x, y) = y/π ∫-∞∞ f(t) dt / ((x - t)2+y2) satisfies uxx + uyy = 0. Homework Equations Leibniz' Rule The Attempt at a Solution I'm not even sure Leibniz' Rule can be applied here since there seems to be a discontinuity in the integrand when x=t and y=0. When I...