Multivariable calculus Definition and 269 Threads

Homework Statement I have the function: f(x,y)=x-y+2x^3/(x^2+y^2) when (x,y) is not equal to (0,0). Otherwise, f(x,y)=0. I need to find the partial derivatives at (0,0). With the use of the definition of the partial derivative as a limit, I get df/dx(0,0)=3 and df/dy(0,0)=-1. However, my...

If I have a scalar function of a variable ##x## I can write the derivative as: ##f'(x)=\frac{df}{dx}##. Now suppose ##x## is no longer a single variable but a vector: ## x=(x^1, x^2, ..., x^n)##. Then of course we have for the derivative ##(\frac{\partial f}{\partial x^1}, ..., \frac{\partial...

I have found that multivariable limits are harder to find and/or prove that something exists. Do you have any recommendations, given questions like "find(if exists) the limit...". For example, I have no idea how to even start thinking about the following limit(if it exists or not, and if it...

Hi, I want to re-learn multivariable calculus, after I have learned it, not in the best possible way... and feel bad about it. I have seen the recommendations here about Hubbard/Shifrin/Fleming/Edwards. I have also seen the books by Munkres/Spivak/Apostol. I didn't really like Hubbard's book...

Homework Statement Hey guys. I am having a little trouble answering this question. I am teaching myself calc 3 and am a little confused here (and thus can't ask a teacher). I need to find the limit as (x,y) approaches (0,1) of f(x,y) when f(x,y)=(xy-x)/(x^2+y^2-2y+1). Homework Equations...

Hello everyone, I have a theoretical calculus question. I am working on a exercise where you have to consider f(x,y,z) and express the variable z as a function of x and y on a certain level surface around a certain (x0,y0,z0). I found out that the condition for this to be able is that the...

Homework Statement Use spherical coordinates to find the volume of the solid enclosed between the spheres $$x^2+y^2+z^2=4$$ and $$x^2+y^2+z^2=4z$$ Homework Equations $$z=\rho cos\phi$$ $$\rho^2=x^2+y^2+z^2$$ $$dxdydz = \rho^2sin\phi d\rho d\phi d\theta$$ The Attempt at a Solution The first...

Suppose you have a parameterized muli-varied function of the from ##F[x(t),y(t),\dot{x}(t),\dot{y}(t)]## and asked to find ##\frac{dF}{dt}##, is this the correct expression according to chain rule? I am confused because of the derivative terms involved. ##\frac{dF}{dt}=\frac{\partial...

Homework Statement I am to prove (using the equations for gradient, divergence and curl in spherical polar coordinates) that vector field $$\mathbf{w}=w_{\psi}(r,\theta)\hat e_{\psi}$$ is solenoidal, find $$w_{\psi}(r,\theta)$$ when it's irrotational and find a potential in this case. Homework...

Hi, I am a math major, and currently studying vector calculus. Since I am feeling that I don't really learn it properly, I am going to re-learn it again in the summer. I would like to improve both my theoretical and computational skills. I am also searching for a book that starts from the...

Homework Statement The cylinder x^2 + y^2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on the ellipse furthest from the origin. Homework Equations $f(x) = x^2 + y^2 + z^2$ $h(x) = x^2 + y^2 = 1$ $g(x) = x + z = 1$ The Attempt at a Solution $\langle 2x, 2y, 2z \rangle...

Homework Statement Let ##T \subset R^3## be a set delimited by the coordinate planes and the surfaces ##y = \sqrt{x}## and ##z = 1-y## in the first octant. Write the intgeral \iiint_T f(x,y,z)dV as iterated integrals in at least 3 different ways. Homework Equations \iiint_T f(x,y,z)dV =...

Hi, I am currently in the first year of my undergraduate mathematics degree and I am taking a course in vector calculus. The course content is: line integrals, conservative field, divergence, gradient, curl, the divergence theorem, green’s formula, Stokes' them., field theory. I have seen that...

Homework Statement The problem asks for a single triple integral (the integrand may be a sum but there must be a single definition for the bounds of the integral) representing the volume (in the first octant) of the shell defined by a sphere of radius 2 centered around the origin and a sphere...

Homework Statement Find the limit \lim_{(x,y)\to(2,2)}\frac{x^3-y^3}{x-y} Homework Equations \epsilon - \delta, baby: If the limit L exists, \forall \: \epsilon \: \exists \: \delta: 0 < \sqrt{(x-a)^2+(y-b)^2} < \delta \rightarrow |f(x,y)-L| < \epsilon The Attempt at a Solution By...

Homework Statement I need to see if the function defined as##f(x,y) = \left\{ \begin{array}{lr} \frac{xy^2}{x^2 + y^2} & (x,y)\neq{}(0,0)\\ 0 & (x,y)=(0,0) \end{array} \right.## is differentiable at (0,0) Homework Equations [/B] A function is differentiable at a...

Hey! I want to do a double integral calculation of this problem##∫∫ xy/(xy^2 +1)^2## over the region bounded by 2 ≤ x ≤ 3 and 2*sqrt(1+x) ≤ y ≤ 2*sqrt(2+4x) on MATLAB and i have tried the following syntax: clc clear all fun=@(x,y) x*y./((x*y.^2+1).^2); ymax=@(x) 2*sqrt(2+4*x)...

The problem is: The temperature (in degrees Celsius) of a metal plate, located in the xy -plane, at any point (x, y ) is given by the function of two variables T(x, y ) = x sin y + y2 sin x. (a) Find the rate of change in temperature in the direction of the positive x-axis at the point (π, π)...

Let us assume the validity of Ampère's circuital law\oint_{\gamma}\mathbf{B}\cdot d\mathbf{x}=\mu_0 I_{\text{linked}}where ##\mathbf{B}## is the magnetic field, ##\gamma## a closed path linking the current of intensity ##I_{\text{linked}}##. All the derivations of the Biot-Savart law for a...

Homework Statement Use the epsilon delta definition to show that lim(x,y) -> (0,0) (x*y^3)/(x^2 + 2y^2) = 0 Homework Equations sqrt(x^2) = |x| <= sqrt(x^2+y^2) ==> |x|/sqrt(x^2+y^2) <= 1 ==> |x|/(x^2+2y^2)? The Attempt at a Solution This limit is true IFF for all values of epsilon > 0, there...

Homework Statement This is a solution to a problem that was on a quiz, and I am confused about how to do it. Especially lines two (<0,1,1>=v(0)=<C1, 1+C2, C3> --> C1=0, C2=0, C3=1 and five (<1,0,0>=r(0)=<1+ K1, K2, K3> -->K1=0, K2=0, K3=0 How do you do these steps...

I have to evaluate $$\int^1_{-1} \int^{ \sqrt {1-x^2}}_{ - \sqrt {1-x^2}} \int^1_{-\sqrt{x^2+y^2}}dzdydx$$ using spherical coordinates. This is what I have come up with $$\int^1_{0} \int^{ 2\pi}_0 \int^{3\pi/4}_{0}r^2\sin\theta d\phi d\theta dr$$ by a combination of sketching and...

I was trying to understand a proof in multivariable calculus (about fundamental pde's), and at some point it said that if ##U## is a convex open set of ##\mathbb{R}^2##, then it's second projection is an open set in ##\mathbb{R}##, without any further explanation. Could you explain please ?

Having a melt down as I have done this problem twice now and my exam is tomorrow and I can't seem to figure it out anymore... ugh. 1. Homework Statement The depth of a lake at the point on the surface with coordinates (x, y ) is given by D(x, y ) = 100−4x 2 −y 2 . a) If a boat at the point (−1...

Say I have a function of three variables, ##F=F(s_{12},s_{23},s_{13}) = F(s,t,-s-t)##, where ##s_{12}=s,s_{23}=t## and ##s_{13}=u = -s-t##. I want to compute the differential operators $$\frac{\partial}{\partial s}, \frac{\partial}{\partial t}\,\,\text{and}\,\,\frac{\partial}{\partial u}.$$...

Homework Statement Find all points at which the direction of fastest change of the function f(x,y) = x^2 + y^2 -2x - 2y is in the direction of <1,1>. Homework Equations <\nabla f = \frac{\delta f}{\delta x} , \frac{\delta f}{\delta y} , \frac{\delta f}{\delta z}> The Attempt at a Solution...

Hi, I'm looking for a good book to study multivariable calculus from that would fit a proof-based undergraduate math course. The emphasis should be on RIGOROUS proofs, theorems and presentation of concepts. I really need a book that would be approachable for self-study with clear and...

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...

Homework Statement Let ##M## be the set of all points ##(x,y) \in \mathbb{R}^2## satisfying the equation ##xy^3 + \frac{x^4}{4} + \frac{y^4}{4} = 1 ## Prove that ##M## is a manifold. What is the dimension of ##M##? Homework EquationsThe Attempt at a Solution I think this question it started...

What are some rigorous theoretical books on mathematics for each branch of it? I have devised a fantastic list of my own and would like to hear your sentiments too. Elementary Algebra: Gelfand's Algebra Gelfand's Functions & Graphs Burnside's Theory of Equations Euler's Analysis of the...

I'm looking for a well written textbook for multivariable calculus textbook. For single variable I worked with Stewart's calculus mostly and I found it very good. However I took a glance at Stewart's multivariable calculus textbook and I didn't enjoy it. The explanations weren't all that good (I...

Homework Statement Let P be the tangent to the graph of g(x,y) = 8-2x^2-3y^2 at the point (1, 2, -6). Let f(x,y) = 4-x^2-y^2. Find the point on the graph of f which has tangent plane parallel to P.Homework Equations g(x,y) = 8-2x^2-3y^2 at (1, 2, -6) f(x,y) = 4-x^2-y^2 The Attempt at a...

I have two problems I need help with 1. Homework Statement https://ccle.ucla.edu/mod/resource/view.php?id=801511 https://ccle.ucla.edu/mod/resource/view.php?id=778704 2,3. Answers and work are givenIn the surface integral problem, I do not understand how it sets it up for Method 1. (How does...

Dear Physics Forum personnel, I am a college sophomore with double majors in mathematics & microbiology and an aspiring analytic number theorist. I will be going to self-study the vector calculus by using Hubbard/Hubbard as a main text and Serge Lang as a supplement to Hubbard; this will help...

Homework Statement Find the volume of the solid. Under the paraboloid z = x^2 + y^2 and above the region bounded by y = x^2 and x = y^2 Well, those curves only intersects in the xy-plane at (0,0) and (1,1), and in the first Quadrant, and in that first Quadrant y = sqrt(x), and over that...

Dear PF personnel, I am a college sophomore with double majors in mathematics and microbiology. I wrote this email to seek your advice on selecting a theoretical, proof-based textbook on the multivariable calculus. I will be taking a multivariable calculus on this Summer but it unfortunately...

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...

I do not know multivariable calculus. I have studied out of Apostol Vol.1. I do not want to learn the material from Apostol Vol. II. Therefore I want to know If it would be worthwhile to go through Hubbard and Hubbard's 'Vector Calculus, Linear Algebra, and Differential Forms' after going...

Hi guys, I have a question about the book "Calculus: An Intuitive and Physical Approach" by Morris Kline. I was wondering if it includes a sufficient coverage of multivariable calculus as well as single variable? I am about to take a course in calculus of several variables and I am a bit...

Let A(a, b, c) and A'(a′,b′,c′) be two distinct points in R3. Let f from [0 , 1] to R3 be defined by f(t) = (1 -t) A + t A'. Instead of calling the component functions of f ,(f1, f2, f3) let us simply write f = (x, y, z). Express x; y; z in terms of the coordinates of A and A, and t. I thought...

Let U={(x,y) in R2:x2+y2<4}, and let f(x,y)=√.(4−x2−y2) Prove that f is differentiable, and find its derivative. I do know how to prove it is differentiable at a specific point in R2, but I could not generalize it to prove it differentiable on R2. Any hint?

Homework Statement $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ Homework Equations Below The Attempt at a Solution $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ I don't understand, we say: $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ Then we say: $$I = \int_{-\infty}^{\infty} e^{-t^2} dt$$...

I am in Calculus BC in high school right now and I am really enjoying it and have finished all of the material for the year and I have heard different things about what class comes after Calculus BC (I believe BC is equivalent to Calc 1 & 2 in college). I have heard the next class, aka Calc 3...

Homework Statement Here it is: Let Ω be a convex region in R2 and let L be a line segment of length ι that connects points on the boundary of Ω. As we move one end of L around the boundary, the other end will also move about this boundary, and the midpoint of L will trace out a curve within Ω...

Recently I started with multivariable calculus; where I have seen concepts like multivariable function, partial derivative, and so on. A week ago we saw the following concept: directional derivative. Ok, I know the math behind this as well as the way to compute the directional derivative through...

This is more a conceptual question. So i am doing some self review of multi variate calculus and i am looking at functinal relations of the form F(x, y, z,...) = 0 In the book they talk about implicit differentiation. Now i fully understand how to do the mechanics of it, but i was trying to...

Okay so I just have a question on triple integrals. I understand how to use triple integrals to find volumes, but what I really don't understand is what I am really getting when I take the triple integral OF a function. I understand physical examples like taking the triple integral of a...

1. The problem I am trying to prove the following relation in cartesian coordinates. We were given a hint to use integration by parts, as well as the fact that we know $d \vec r = dx\,dy\,dz$ (volume integral). $$\int f(\vec r)\ \nabla \cdot \vec A(\vec r) \, d \vec r = -\int \vec A(\vec...

Homework Statement Let R be the solid region that is bounded by two spheres x^2 + y^2 + z^2=1 and x^2 + y^2 + z^2=2. Determine the moment of inertia of R around the x-axis if the mass density per unit volume of R is u=sqrt(x^2 + y^2 + z^2). Homework Equations Moment of Inertia around the...

Homework Statement Let the path C traverse part of the circle or radius 3 at the origin, in a clockwise direction, from (0,-3) to (3,0). Calculate the total mass of a wire in shape C, if the mass density of the wire is u=x^2+4y Homework Equations mass of plate equation= double integral u(x,y)...