Multivariable Definition and 524 Threads

Homework Statement Determine whether the limit exists; if it does, what is it? Homework Equations take the limit as (x,y) -> (0,0) of f(x,y) where f(x,y) = (x^6-y^6)/(x^3-y^3) The Attempt at a Solution What i started doing was approaching along the line y=0 and that would give lim as...

Homework Statement Show that \nabla(r^n)=nr^(n-2)r if n is a position integer. (hint:use \nabla(fg)=f\nablag+g\nablaf) Homework Equations let r(x,y,z) = xi+yJ+zK be the position vector and let r(x,y,z)= |r(x,y,z)| The Attempt at a Solution I tried separating \nabla(r^n) to...

Homework Statement I was given an assignment to find all relative extrema and saddle points of the equation f(x,y) = 1/3x^4 + 1/2y^4 - 4xy^2 +2x^2 + 2y^2 + 3 I derived the first partial with respect to x and the first partial with respect to y, but when I tried to find where they both...

Hello hello. In class we went over the ''mini-chain rule'' once, and haven't gone over the real chain rule yet. I really want to understand how to go about solving this equation, and to really understand what is happening here. x=u3-3uv2 y=3u2v-v3 z=u2-v2 Define z as a function of x and...

Hello, if I have the following unknown function f(x_1,\ldots,x_n) Assuming I am given all its partial derivatives \frac{\partial f}{\partial x_i} is it possible to get the original function f ? This is clearly possible for a one-variable function f(x). If we know df/dx we just need to...

I am trying to find the second derivative of the function C:[0,1]^{2} \rightarrow [0,1] ,\quad \mbox{defined by }C=C(u,v) evaluated at u=F(x)=1-\exp(-\lambda_{1} x),\quad \lambda_{1} \geq 0 and v=G(x)=1-\exp(-\lambda_{2} x),\quad \lambda_{2} \geq 0 First I work out the first...

Ok, so I am doing a typical momentum problem with a varying mass (here dm/dt > 0). I am also taking into account air resistance (stoke's law essentially). After some manipulations I basically get: 6πnRv - mg(1 - p(water)/p(object)) = m (d/dt(v)) + v (d/dt(m)) I know I need to separate the...

I've got a problem where I need to numerically integrate a multivariable function but I'm not sure how to do this. I'm more than familiar with how to numerically integrate a single variable function numerically but I have no clue how to do this for a multivariable function. For example let's say...

Hi, i was wondering if someone could please help to find and classify the critical points of : f(x,y) = (x-y)^2 What i know: I got fx = 2(x-y) and fy = -2(x-y) and in order to find the critical points we need to solve: 2(x-y) =0 -2(x-y) = 0 so if x =y then the above hold. where...

As mentioned in the title~ Could anyone give me a hint or an idea ? Thanks~

lim (x^2-4y^2)/(x+2y) as (x,y)->(-2,1) Does this limit go to -4 or DNE since it is undefined all along x+2y?

Hi, Kinda need help for this question. f(x,y) = ln(y-2x) 1. Find the largest possible domain 2. Find the range of the function where the function is defined over the largest possible domain 3. find the largest possible domain if it is desired that f(x,y) > 0 Explain why two...

1.) Homework Statement Sketch the domain of f(x,y,z) = ln(25-x^2-y^2-z^2) and determine it's range. Homework Equations N/a The Attempt at a Solution Im sure the domain is x^2+y^2+z^2 < 25 or (-infinity, 25) Then the range is (0,+infinity.) my problem is how would i sketch it.. i was...

Homework Statement Verify the following limit by using delta-epsilon arguments Homework Equations lim (x, y) -> (1, -1) of xy^2 = 1 The Attempt at a Solution Right, so I'm having some trouble with these delta-epsilon proofs for multivariable limits. Some of them are easier than others...

Consider a (possibly complex-valued) function F(z) = F(x,y) of two variables. Can it make sense to talk about the differential dF of this function without it having a derivative dF/dz? Or must F be differentiable before we can even start talking about dF?

Application of a Limit of a multivariable function..HELP! Homework Statement if f (x,y) = (xy^2)/(x+y^2). prove that for every real number a there is a path along which f (x,y) will approache a, as (x,y) is approaching (0,0). Homework Equations f (x,y) = (xy^2)/(x+y^2) A as a...

Homework Statement Asked to find the limit of [6*(x^3)*(y^2)] / [2*(x^4) + (y^4)] as (x,y) is approaching (0,0) Homework Equations lim, as (x,y) ---> (0,0), of [6*(x^3)*(y^2)] / [2*(x^4) + (y^4)] x = rcos(theta); y = rsin(theta) The Attempt at a Solution Tried numerous times...

Homework Statement the limit as x,y go to 0,0 of (x^2*(sin(y))^2) / (x^2 + 2y^2) The Attempt at a Solution I can see that the limit goes to 0 when x=y, and when y=0 the limit goes to 1. Is this proof that the limit does not exist? I'm trying to work ahead and learn straight from...

I'm taking multi-variable after having a while off from school, so forgive me if these are simple ones that I just don't "see" Homework Statement lim (x, y) --> 0, 0 \frac{x^2 y^2 e^y}{x^4+4y^2} and lim (x, y) --> (1, 1) \frac{x-y}{x^3-y} Homework Equations The Attempt at a Solution The...

1. A medieval city has the shape of a square and is protected by walls with length 550 m and height 16 m. You are the commander of an attacking army and the closest you can get to the wall is 100 m. Your plan is to set fire to the city by catapulting heated rocks over the wall (with an initial...

If there was a function f(x,y) = z...would the inverse be defined under injection (one-to-one correspondence) only if this condition was held?: Every z corresponds to only one (x,y) pair. If this was the case...then wouldn't that imply that a good number of 2-D surfaces cannot have an inverse...

Help with basic multivariable problem. [SOLVED] Homework Statement Two surfaces intersect at a space curve C. The two surfaces are 4y2 + 9z2 = 36 and x = 2y2 - 3z2 Find a vector parametrization for C. (r(t) = ( f(t) , g(t) , h(t) ) Find a set of values for the parameter t over which C is...

Hi everybody. I'm currently taking Calculus III with applications, and the book they gave us was Multivariable Calculus by Ron Larson. I wanted to Calc III, which is more pure math as opposed to the class I'm in that's mostly for engineers (theres a third class oriented even more for...

1. Write inequalities to describe the region: The solid cylinder that lies on or below the plane z=8 and on or above the disk in the xy-plane with a center the origin and radius 2. I don't understand because I'm using stewart's calculus i have no idea what the equation for a cylinder is...

Homework Statement I got a few functions I need to expand to series using Maclaurin forumlas. Homework Equations http://mathworld.wolfram.com/MaclaurinSeries.html The Attempt at a Solution So here are the ones I managed to do: f= \sqrt{1-x^2-y^2} writing it in another form...

Hello all, I am stuck on what seems like a rather simple problem: Let f:\mathbb{R}^3 \rightarrow \mathbb{R} and g:\mathbb{R}^2\rightarrow \mathbb{R} be differentiable. Let F:\mathbb{R}^2 \rightarrow \mathbb{R} be defined by the equation F(x,y)=f(x,y,g(x,y)). Find DF in terms of the...

Does anyone have some suggestions for a good multi-variable/vector calculus book? I have a fairly reasonable math background - managed to self-teach myself calculus through Micahael Spivak's text Calculus pretty successfully - and I'm looking for something that's fairly rigorous. One of my...

I've been stuck on this problem for quite a while now and could use some assistance: Find the limit (or prove that it does not exist): lim{(x,y)->(1^+,oo)} x^(-y) I've tried switching to polar and end up with y=rsin(@) implying r diverges, which implies cos(@) must tend to zero for x...

Homework Statement Find the average area of an inscribed triangle in the unit circle. Assume that each vertex of the triangle is equally likely to be at any point of the unit circle and that the location of one vertex does not affect the likelihood the location of another in any way. (Note...

Find the average area of an inscribed triangle in the unit circle. Assume that each vertex of the triangle is equally likely to be at any point of the unit circle and that the location of one vertex does not affect the likelihood the location of another in any way. (Note that, as seen in Problem...

[SIZE="5"]The Problem: Problem is reference to Problem 23 on Page 210 of Basic Multivariable Calculus by Marsden, Tromba, Weinstein. But my problem is a bit different. The main problem is: For the function f(x,y) = y2+3x4-4x2y , is (0,0) a local maxima, minima or saddle point? The...

Homework Statement Find the limit of: (x^2+y^2+2xy)/(x^2+y^2)Homework Equations x = r*cos(theta) y= r*sin(theta) The Attempt at a Solution So what I did was change to polar coordinates. Then it simplifies to: (r^2 + 2r^2cos(theta)sin(theta) )/r^2 Factoring out an r^2 from everything you...

hi, my calc class' designated textbook is calculus:several variables by robert a. adams. however, stewarts' multivariable calculus is also recommended. which one would you recommend out of the two?

let g:[a,b] -> R be a function that is continuous almost everywhere. assume that g(x) > 0 on [a,b]. Show that the set S = { (x,y): 0 <= y <= g(x) , a <= x <= b} is rectifiable. One way to attack it, is to show that S is bounded and boundary of S has measure zero. the problem I am having is...

Homework Statement I would like to be able to compute multivariable limits with a CAS. I have Mathematica, Derive and an HP 50g. For example, how would I compute: \lim_{(x,y)\rightarrow(0,0)}\frac{x^2}{y} Homework Equations The Attempt at a Solution On Mathematica I tried: Limit[x^2/y, {x, y}...

Homework Statement lim (x,y)->(0,0) (cos(x) - 1 + (x^2/2)) / ( x^4 + y^4) The Attempt at a Solution Am I correct in assuming that I have to solve by looking for a path where the limit doesn't exist/is different to another path (given that the epsilon thing is beyond the scope of the course)...

whats a good book to learn multivariable and vector calculus in a rigorous fashion, similar to spivak. id prefer the vector calculus to stick to R3 (ie stokes), ill save the n case for analysis. apostol II looks like a good book, but its very expensive and i generally don't like apostol...

i'm going into the second semester of my sophomore year. I'm a math major, looking to go to grad school for pure math. i was just wondering my chances of getting into an reu this summer (anywhere/anytype) including this upcoming semester i will have taken: -multivariable calc -*advanced...

Homework Statement Let f: R^n -> R be defined as follows: f(x) = x*L(x) where * denotes the standard inner product and L: R^n -> R^n is a linear function. I'm trying to find the directional derivative f'(x;u). Homework Equations I know that f'(x;u) (the directional derivative of...

How do I go about finding the limit of a multivariable function? Example: limit as (x,y) approach (0,0) of: (x + 2y) / sqrt (x^2 + 4(y^2)) Do I need to use partial derivatives?

I was reading a statistics book, and part of the problem reduces to the calculus problem of doing the following: 1) Let u=x/y, v=y, with domain 0<x<y<1[/color], how to find the ranges of u and v after the transformation? 2) Let u=x/(x+y), v=x+y with domain x>1, y>1[/color], what values...

Homework Statement I need to show that |sin(e^xy)-sin(1)|/(x^2+y^2)^1/2 -> 0 as (x,y) -> (0,0) Homework Equations Triangle Inequality? The Attempt at a Solution I know that this is true, since e^xy -> 1 as (x,y) -> (0,0) much, much faster than (x^2+y^2)^1/2 -> 0 as (x,y) -> (0,0)...

Homework Statement Let f(x,y)=2x+3y. Let \epsilon be any positive number. Show that there is a disk with center (1,1) such that whenever P is in that disk, |f(P)-5|< \epsilon. (Give \delta as a function of \epsilon.) Homework Equations None. The Attempt at a Solution Um, I tried...

Not really a homework problem... just a general question (this seemed like the place to put it...). Say I have three functions: f,g,h:\mathbb{R}^2\rightarrow\mathbb{R}^3 and an expression along the lines of: \left\langle f(u_1,u_2),g(u_1,u_2)\right\rangle h(u_1,u_2) What...

Homework Statement lim(x,y)->(4,pi)(x^2sin(x/y)), i tried it by polar as well as epsilon delta method, but by polar coordinate method it is coming that " limit doesn't exits" but by epsilon delta method "it exists & is equal to (epsilon+(16/(2)^(1/2)) )^(1/2)-4. Homework...

(Keep in mind, I made this off the top of my head, so if something cancels easy, ignore it) Let's say I had this expression: f(x,y)=\frac{y^2-xy+1}{(x+y)(x-y)} I want to decompose this to: \frac{A}{x+y} + \frac{B}{x-y} So i begin the process: y^2-xy+1=A(x-y) + B(x+y) y^2-xy+1=x(A+B) +...

Homework Statement Show that among all parallelograms with perimeter l, a square with sides of length l/4 has maximum area. Do this using the second partials test, and then using Lagrange multipliers. Homework Equations Area of a parallelogram: A = absin\phi, where a and b are the...

Hi guys... Haven't been in the forum for a couple years now. I have an old analysis problem that I never manage to solve. Would be nice if someone can shed some light on this.let f be a C^1 function from \mathbb{R}^n \rightarrow \mathbb{R}^n, n>1. df is invertible except at isolated points (WLOG...

I can't get this problem right and it's part of a web-based assignment that I have to submit. In order to get credit for the problem, all of my answers need to be right; I've tried many different times and I can't seem to figure out what I am doing wrong. Here it is with my explanations as to...

Homework Statement For: f(x,y)=(x+y)/(x^2+2y^2+6) Find the stationary points of f Homework Equations The Attempt at a Solution To find a stationary point the first partial derivative must equal zero, correct? I've found the first partial derivatives using the quotient...