What is Multivariable calculus: Definition and 273 Discussions

I am studying Analysis on Manifolds by Munkres. He introduces improper/extended integrals over open set the following way: Let A be an open set in R^n; let f : A -> R be a continuous function. If f is non-negative on A, we define the (extended) integral of f over A, as the supremum of all the...

Hello. I am studying Analysis on Manifolds by Munkres. I have a problem with a proof in section 20. It states that: Let A be an n by n matrix. Let h:R^n->R^n be the linear transformation h(x)=A x. Let S be a rectifiable set (the boundary of S BdS has measure 0) in R^n. Then v(h(S))=|detA|v(S)...

Homework Statement Let ##x,y \in \mathbb{R^n}## not null vectors. If for all ##z \in \mathbb{R^n}## that is orthogonal to ##x## we have that ##z## is also orthogonal to ##y##, prove that ##x## and ##y## are multiple of each other. Homework Equations We can use that fact that ##<x ...

Homework Statement The path integral from (0,0,0) to (1,1,1) of $$<x^2,2yz,y^2>$$. I am a little confused about the setup.Homework Equations $$\int_{a}^{b} v.dl$$The Attempt at a Solution Here is how I set it up. $$\int_{0}^{1}x^2 dx + \int_{0}^{1}2yz dy + \int_{0}^{1}y^2 dz$$ Since the...

Homework Statement Let ##a > 0##. Find the mass of the "solid bowl" consisting of points inside the paraboloid ##z=a(x^2+y^2) \text { for } 0\leq z \leq H \text{. Assume a mass density of } \rho(x, y, z) = z##. Homework Equations ##x^2 + y^2 = z^2## The Attempt at a Solution [/B] mass = ##m...

Homework Statement Homework EquationsThe Attempt at a Solution I have attached the problem and solution. I don’t know how to do part b even I have looked at the solution. How to transform the original cartesian equation to the semi cylindral coordinate equation? Is there is systematical...

Homework Statement Given $$\phi = x^{2} +y^{2}-z^{2}-1 $$ Calculate the unit normal to level surface φ = 0 at the point r = (0,1,0) Homework Equations $$ \hat{\mathbf n} = \frac{∇\phi}{|\phi|}$$ $$ z = \sqrt{x^{2}+y^{2} -1} $$ $$ \mathbf n = (1,0,(\frac{\partial z}{\partial x})_{P})...

Homework Statement I'm working on a generalization of gravitation to n dimensions. I'm trying to compute gravitational attraction experienced by a point mass y due to a uniform mass distribution throughout a ball of radius a -- B(0, a). Homework Equations 3. The Attempt at a Solution [/B]...

Let $h$ be a bump function that is $0$ outside $B_\epsilon^m(0)$ and posetive on its interior. Let $f$ be smooth function on $B_{2\epsilon}^m(0)$. Define $f^*(x)=h(x)f(x)$ if $x\in B_{2\epsilon}^m(0)$ and $=0$ if $x\in \mathbb{R^m}-B_\epsilon^m(0)$. I want to show that $f^*$ is smooth on...

So, I'm now studying thermodynamics and our teacher proved some time ago the following mathematical result: If f(x,y,z)=0, then (∂x/∂y)z=1/(∂y/∂x)z But today he used this relation for a function of four variables. Does this result still hold, because I'm not really sure how to prove it. If...

calculate the work done by the force field $F(x,y)=(ye^{xy})i+(1+xe^{xy})j$ by moving a particle along the curve C described by gamma (γ):[0,1] in $R^2$, where gamma (γ)=(2t-1, t²-t)

(I) Find the limit (x,y)->(0,0) of F, then prove it by definition. (II) Find the limit and prove it by definition of: as (x,y) approach (C,0), C different from zero. I have previously asked it on Quora, but it doesn't appear to have answers any...

Homework Statement Griffiths Introduction to Electrodynamics 4th Edition Example 1.10 Check the divergence theorem using the function: v = y^2 (i) + (2xy + z^2) (j) + (2yz) (k) and a unit cube at the origin. Homework Equations (closed)∫v⋅da = ∫∇⋅vdV The flux of vector v at the boundary of the...

Consider a double integral $$K= \int_{-a}^a \int_{-b}^b \frac{B}{r_1(y,z)r_2^2(y,z)} \sin(kr_1+kr_2) \,dy\,dz$$ where $$r_1 =\sqrt{A^2+y^2+z^2}$$ $$r_2=\sqrt{B^2+(C-y)^2+z^2} $$ Now consider a function: $$C = C(a,b,k,A,B)$$ I want to find the function C such that K is maximized. In other...

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 -...

What are the math concepts I have to learn for Radiometry, Photometry and Thermodynamics (all Calculus-based) as applied in building science (engineering, architecture, etc.). I'm almost done with Multivariable Calculus and I'm aware that MV Calculus is necessary, but what specific calculus...

1. Find if the limit exist: sin (x^3 + y^3) / (x + y) (x,y)-> (0,0) So I am starting solving this by using polar coordinates form and I get to lim= sin r^3 ( cos^3θ + sin^3θ) / r ( cosθ + sinθ) = lim r^2 ( cos^2Θ + sin^2Θ) My question is ok so far and how...

Hi, friends! Under particular conditions on ##\phi:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}## - I think, as said here, that it is sufficient that ##\phi\in C_c^1(\mathbb{R}^4)##: please correct me if I am wrong - the following equality holds$$\frac{\partial}{\partial r_k}\int_{\mathbb{R}^3}...

Books on multivariable calculus that I often see get good recommendations are, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard and Hubbard Vector Calculus by Colley What are other good books with some material on differential forms like Hubbard and Colley? Books by Edwards...

Hello, friends! I know, thanks to @Hawkeye18 who proved this identity to me, that, if ##\phi:V\to\mathbb{R}## is a bounded measurable function defined on the bounded measurable domain ##V\subset\mathbb{R}^3##, then, for any ##k\in\{1,2,3\}##, $$\frac{\partial}{\partial r_k}\int_V...

Hi guys, i´m pretty well in calculus 1 and i´m studying for the International Physics Olympiad. So I´d like to know some multivariable calculus books that cover vector calc too, are balanced (proofs are welcome) and emphasizes physical intuitions. Thank you already!

I am almost on the verge of completing single-variable Calculus, and I've got a book on the same by I. A. Maron. So, after getting a good grip on single-variable Calculus, I want to start with multivariable. Can anyone recommend me good books on multivariable Calculus with which I could begin...

Homework Statement Find any maxima/minima on f(x,y) = x2+2y2 on the unit circle, centered at the origin. Homework Equations grad f = λgrad g constraint: 1=x2+y2 The Attempt at a Solution grad f = 2xi+4yj grad g = 2xi+2yj 2x=λ2x 2y=λ4y How do I solve this? I don't see any way to get numbers...

Hi, I'm asking for a friend who will be majoring in chemical engineering. We have already taken Calculus I, II, and III under a course offered by a local community college. Admittedly, it was taught from stewart's series of calculus books, and we did exactly zero proofs in the class, and all...

I am a student currently taking both Multivariable Calculus and Differential Equations. Instead of a final exam my teacher assigned a final project for Multivariable, and I chose to do something with Spacetime/Black holes. Within the scope of <100 hours of work, is there anything I can do with...

I am looking at a proof from a book in fluid dynamics on time differentiation of fluid line integrals - Basically I am looking at the second term on the RHS in this equation $$ d/dt \int_L dr.A = \int_L dr. \partial A / \partial t + d/dt \int_L dr.A$$ The author has a field vector A for a...

Hi All, $$\int{\exp((x_2-x_1)^2+k_1x_1+k_2x_2)dx_1dx_2}$$ I can perform the integration of the integral above easily by changing the variable $$u=x_2+x_1\\ v=x_2-x_1$$ Of course first computing the Jacobian, and integrating over ##u## and ##v## I am wondering how you perform the change of...

Hello. I am having a lot of trouble trying to solve/analyse this integral: $$\displaystyle \int_2^\infty \frac{x+y}{(y)(y^2-1)(\ln(x+y))} dy$$ I have tried everything with no result; it seems impossible for me to work with that natural logarithn. I have also tried to compute it, as it...

Homework Statement A vector field $\vec F$ is defined in cylindrical polar coordinates $\rho , \theta , z$ by $\vec F = F_0(\frac{xcos (\lambda z)}{a}\hat i \ + \frac{ycos(\lambda z)}{a}\hat j \ + sin(\lambda z)\hat k) \ \equiv \frac{F_0 \rho}{a}cos(\lambda z)\hat \rho \ + F_0sin(\lambda...

Homework Statement Compute the work of the vector field $$H: \mathbb{R^2} \setminus{(0,0}) \to \mathbb{R}$$ $$H(x,y)=\bigg(y^2-\frac{y}{x^2+y^2},1+2xy+\frac{x}{x^2+y^2}\bigg)$$ in the path $$g(t) = (1-t^2, t^2+t-1)$ with $t\in[-1,1]$$ Homework Equations 3. The Attempt at a Solution [/B] So...

Homework Statement $$f:\mathbb{R^2}\to\mathbb{R}$$ a differentiable function in the origin so: $$f(t,t) =t^3+t$$ and $$f(t,-2t)=2t$$ Calculate $$D_vf(0,0)$$ $$v=(1,3)$$ Homework Equations 3. The Attempt at a Solution [/B] I have no idea on how to approach this problem. I know that...

Homework Statement Compute the work of the vector field ##F(x,y)=(\frac{y}{x^2+y^2},\frac{-x}{x^2+y^2})## in the line segment that goes from (0,1) to (1,0). Homework Equations 3. The Attempt at a Solution [/B] My attempt (please let me know if there is an easier way to do this) I applied...

Homework Statement Let f(x, y) = x^2 + kxy + y^2 , where k is some constant in R. i. Show that f has a stationary point at (0, 0) for every k ∈ R Homework Equations ... The Attempt at a Solution I may have the solution or i may have gone completely wrong I am not entirely sure. i first found...

Homework Statement f(x,y) = 1/y^2-x find the domain of f. Given c ∈ R \ {0} find (x, y) ∈ R 2 such that f(x, y) = c. Finally determine the range of f. Homework Equations I know that the domain of the function is anywhere that the function is defined. The Attempt at a Solution in the case of...

Homework Statement If f(x) is a scalar-valued function, show that ∂ƒ²/∂xi∂xj are the components of a Cartesian tensor of rank 2. Homework Equations N/A The Attempt at a Solution I don't even know where to begin. We began learning tensors in multivariable calculus (though I don't think this is...

A simple method to find the potential of a conservative vector field defined on a domain ##D## is to calculate the integral $$U(x,y,z)=\int_{\gamma} F \cdot ds$$ On a curve ##\gamma## that is made of segments parallel to the coordinate axes, that start from a chosen point ##(x_0,y_0,z_0)##. I...

Hello, friends! My textbook, Gettys's Physics, says that the Lorenz gauge choice uses the magnetic vector potential $$\mathbf{A}(\mathbf{x},t):=\frac{\mu_0}{4\pi}\int \frac{\mathbf{J}(\mathbf{y},t-c^{-1}\|\mathbf{x}-\mathbf{y}\|)}{\|\mathbf{x}-\mathbf{y}\|}d^3y $$and the electric potential...

Homework Statement On a sample midterm for my Calc 3 class the following question appears: Find the mass of (and sketch) the region E with density ##\rho = ky## bounded by the 'cylinder' ##y =\sin x## and the planes ##z=1-y, z=0, x=0## for ##0\le x\le\pi/2##. Homework Equations $$ m= \int_{E}...

Homework Statement r(t) = < 2e^t - 5 , e^t +3t^2 , 4t^2 +1> Is a curve that lies within a plane. Find the equation of this plane. The Attempt at a Solution I am not sure if my approach is correct. These are my results: x=2e^t - 5 y = e^t +3t^2 z = 4t^2 + 1 z = 4t^2 + 1...

I am having trouble doing this problem from my textbook... and have no idea how to doit. 1. Homework Statement I am having trouble doing this problem from my textbook... Show that the equation x + y - z + cos(xyz) = 0 can be solved for z = g(x,y) near the origin. Find dg/dx and dg/dy (dg/dx...

Vector fields confuses me. What are the differences between (##t## could be any variable, not just time): 1. If the position vector don't have an argument, ##\mathbf{r}=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat e}_z=(x,y,z)## so ##\mathbf{E}(\mathbf{r},t)=E_x(\mathbf{r},t)\mathbf{\hat...

I am trying to calculate the magnetic field generated by an ideal toroidal solenoid by using the integral of the Biot-Savart law. I do not intend to use Ampère's circuital law. Let ##I## be the intensity of the current flowing in each of the ##N## loops of the solenoid, which I will consider an...

My text of physics, Gettys's, proves that the magnetic field on the axis of a solenoid, in whose loops, of linear density ##n## (i.e. there are ##n## loops per length unit), a current of intensity ##I## flows, has the same direction as the loops' moment of magnetic dipole and magnitude ##\mu_0...

I had posted a question earlier which this is related to, but a different equation. $$\frac{d}{dt} \int_0^t H(t,s)ds = H(t,t) + \int_0^t \frac{\partial H}{\partial t}(t,s)ds$$ This was another formula needed in a proof however I don't see how this one holds either. I tried following a proof of...

Hello, do you know of any books similar in style to Callahan's Advanced Calculus book(a book that explains the geometrical intuition behind the math)? This goes for any subject in mathematics(but especially for subjects like vector calculus, differential geometry, topology). Thanks in advance!

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...