Multivariable Definition and 524 Threads

I have provided my solution here. I just want to be sure if it's correct or not.

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...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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 +...

E.g. all real numbers could be a domain but not necessarily, etc. Am I right?

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...

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?

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...

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...

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...

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.

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...

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...

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...

(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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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) -...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

How do I know where to put the axes for the equation 4x^2 - 9y^2 = z when graphing in 3d?

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...

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...

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...

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?

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...

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...