Multivariable Definition and 536 Threads

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, 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, what values can u and v take...

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...

Homework Statement Suppose that the equation G(s, t, u) = 0 implicitly defines each of the three variables s, t, and u as functions of the other two: s = x(t, u), t = y(s, u), and u = z(s, t). If G is differentiable and Gs, Gt, and Gu are all nonzero, show that 1 = - (δu/δs) · (δs/δt) ·...

(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...

Hey guys here, I am going to university soon. Introduction to Multivariable is one of the courses. Before going to university, I hope to do some self-study as I am relatively weak in math. Hope that with an extra prepared Math module I can cope well during my university's life. May you...

Homework Statement Since both my questions are on the same topic, i'll throw them both in here 1. Find dz/dt for z=(x^2)(t^2), x^2+3xt+2t^2=1 2. Show that if u=xy, v=xy and z=f(u,v) then: x.dz/dx-y.dz/dy=(x-y)dz/dv Homework Equations The Attempt at a Solution 1. I only...

Homework Statement Find the steepest path up a hill as an expression in terms of x and y whose height is given by f(x,y)=x^2y-2xy+5 starting at (2,1) 2. The attempt at a solution I know that I need to get a set of parametric equations for x and y because I have done problems sort of...

Homework Statement Since consumers cannot be outside the set of affordable bundles, we get the rate of exchange that consumers face provided the spend all of their budget dy/dx = -Px/Py (where Px is the price of good x, PY is the price of good y) by totally differentiating the budget...

can someone link/show me a formal proof of the multivariable chain rule?

I think this is the first time I've used this forum for myself. :approve: OK, I'm picking out courses for next semester. Right now I'm in the second semester of Complex Analysis (based on Serge Lang's book) which is a grad level course in single variable complex analysis. My school offers a...

Homework Statement find local max min and saddle point. f(x,y)= sin(x)sin(y), -pi<x<pi, -pi<y<piHomework Equations noneThe Attempt at a Solution fx = cos(x)sin(y) fy= sin(x)cos(y) now how do I get the critical points, I know how to get max min and saddle point, but I don't know how to get...

Homework Statement I can find on Wikipedia the "formula" for integration by parts for the case where there is a multi-variable integrand, but I would like to know what substitutions to make in order to show my steps. Homework Equations For multiple variables we have...

Homework Statement lim (x^2+y^2)/((root(x^2+y^2+1) - 1) (x,y)-->(0,0) what is the limit Homework Equations none The Attempt at a Solution \lim_{(x,y) \to (0,0)} \frac {x^2 + y^2}{\sqrt{x^2 + y^2 + 1} - 1} = \lim_{r \to 0} \frac {r^2}{\sqrt{r^2 + 1} - 1} I got this far the...

Anyone familiar with dot products of two vectors? What does the dot product show, in other words what is the point of doing a dot product?

Hi, everyone- I have a quick question. When you solve for the derivative (as a linear transformation) using the limit definition of derivative, how does it go? For example, let p_k be defined as the projection function from Rn to R, projecting onto kth coordinate of the input value...

I am taking a vector calculus course and am having trouble figuring out how to graph things in Matlab to verify my answers. Here is exactly what I want to do: I want to graph side by side two different graphs simultaneously, so I can compare them. I will give you the equations of the two...

Homework Statement Part 1: A 160lb man carries a 25lb paint can up a spiral staircase, which has radius 20 feet, completes 3 revolutions, and has final height 90 feet. What is the work done? Part 2: This time, the man's paint can leaks at a constant rate such that he loses 9lbs of paint...

I need a refresher on my multivariable differential calculus. Does anyone know of something brief with lots of exercises (maybe something geared towards physicists)?

Homework Statement Maximize the Riemann Sum ##\Sigma^{n}_{i=1}x_{i}*y_i## subject to constraints ##\Sigma^{n}_{i=1}x^{2}_{i}=1## and ##\Sigma^{n}_{i=1}y^{2}_{i}=1 ## Homework Equations My teacher doesn't speak English very well. I'm in Calculus 3 and the average on his exams are...

Ok, I'm pretty much at whit's end trying to figure this review question out. Apparently my teacher forgot to mention that our book couldn't teach us everything we need to know for our test... Anyhow, the question is as follows, and I'm utterly at a loss as to what the answer is: Find the...

Hi, I have a problem in solving a multivariable equation. This multivariable consists of several variable which is known and are insert as input intially. For example: f(x)= a*x+b*x^2+-c*x+d(x^4+e)^2, f(x)=0 and a,b,c,d,e are inputs. I wanted to get the answer x. I have tried to use fzero and...

I am having a terrible hard time with the multivariable chain rule and its related stuff (I read my textbook many times, but it doesn't help that much because the explanations are very limited). I hope that someone can help me to withdraw from this darkness of confusion. 1) (Differentiation...

Homework Statement Find the limit, if it exists, or show that the limit does not exist. limit (x,y) --> (0,0) a) f(x,y) = (xycosy) / (3x^2 + y^2) b) f(x,y) = (xy) / sqrt(x^2 + y^2) c) f(x,y) = ((x^2)ye^y) / ((x^4) + 4y^2) Homework Equations The Attempt at a...

Homework Statement Sketch the following curves on separate number planes. y= 2x^2 x^2 + 2y^2= 4 Homework Equations The Attempt at a Solution For the first one, would the intersection of the curve with the y-axis be at y=2 ( I have attached a diagram of my solution). The...

Homework Statement \lim_{(x,y) \rightarrow (0,0)} \frac{x^2+sin^2 y}{2x^2+y^2} Homework Equations The Attempt at a Solution Since I can't evaluate the limit directly and I can't see a way to get a 0 on the top in order to get 2 different limits I split up the limit into these 2...

I was trying to solve the practice problems in my textbook, but I am highly frustrated. The terrible thing is that my textbook has a few to no examples at all, just a bunch of theorems and definitions, so I have no idea how to solve real problems...I am feeling desperate... Note: Let x E...

[Question] Hi, my teacher gave us this problem, and he couldn't figure out why method was incorrect and why I got the answer I did. Given we know the gradient slope = <-56,1.886> at the point (2,0) on a surface f(x,y), in what direction, expressed as a unit vector, is f increasing most...

Let L be the line through the point Po(1,2,8) which is parallel to the vector R(3i-j-4k). Find the point at which l intersects the plane through the point p1(-4,0,3) having normal vector n(3i-2j+6k) I did the following: x=1+3t y=2-t z=8-4t 3(x+4)+6(z-3)=0 3x+6z-6=0 3(1+3t)+6(8-4t)-6=0...

Note: <= means less than or equal to, >= means greater than or equal to 1) Prove formally that lim xy^2 / (x^2 + y^2) = 0 (x,y)->(0,0) without breaking the vectors into components. Let X=(x,y). We want to prove that for all epsilon>0, there exists a delta>0 such that if...

1) lim [x(y^2)] / (x^2 + y^2) (x,y)->(0,0) Find the value of the given limit, if it exists. Using polar coordinates, set x = r cos(theta), y = r sin(theta) Then, the given limit = lim [r cos(theta) r^2 sin^2(theta)] / r^2 r->0 = lim r [cos(theta)...

hey everybody, I'm currently using the book Multivariable Calculus by james stewart 6E and i have to say... it SUCKS. SUCKS just like his single variable calculus book that we are forced to by at university. So I really need a good book, which one would you guys recommend?

I have no luck with proofs... Prove that B_{r} ((x_{0}, y_{0})) = {(x,y) : || (x,y) - (x_{0}, y_{0})|| < r} is an open set in R. Now I know that to be an open set if and only if each of its points is an interior point and if it contains no boundary points. I would consider trying to prove...

I am repeting some multivariable calculus. I want to know if have done right now: \mathbf{r} = \mathbf{r}(q_1, q_2, q_3) \dfrac{\partial \mathbf{r}}{\partial q_1} = \left(\dfrac{\partial r_1}{\partial q_1} , \dfrac{\partial r_2}{\partial q_1} , \dfrac{\partial r_3}{\partial q_1}...

I'm studying calculus III topics on my own, but I've seen this notation prop up a lot. Could you tell me what it means? http://en.wikipedia.org/wiki/Exact_differential The notation on this page, it has this notation: ( \frac{dA}{dx} )_y = ( \frac{dB}{dy} )_x What do the x's and y's...

Calculus, Stewart - "Calculus" both Single & Multivariable? There are are several different texts so I'm confused. I have "Calculus", does it include both Single & Multivariable?

Find the equation of reflection of the sphere in (x-1)^2+(y+2)^2+(z-4)^2=16 with respect to the point (2,1,-2). There was another question asking for the reflection equation but it was with respect to the xy-plane so it just meant changing some signs. What I came up with for this question was...

Homework Statement The height of a hill is given as the following: h \left( x,y \right) =40\, (\left( 4+{x}^{2}+3\,{y}^{2} \right) ^{-1}) There's a stream passes the point (1,1,5) which is on the surface of h. The stream follows the steepest descent. Find the equation of the stream...

Hi I'm studying for a calculus exam and I'm a little stuck on finding the extrema for multivariable functions. For the particular question I'm trying to do now I need to find and classify the extrema for the function f(x,y) = (4x^2)(e^y) - 2x^4 - e^4y. I can find the first derivatives...

a) Find the area of the part of the surface S = {x^2+ y^2+ (z-1)^2 = 4, 0 ≤ z ≤ 1}. Note that this is part of the sphere of radius 2 with center (0,0,1).

hey, hope this is the right spot for this... im curious to know what Domain and Range refer to in a multvariable function. I understand what it means in a single variable (x is usually domain, y is usually range), but when it gets to Multivariable, it doesn't make sense to me. Domain...

Ok there's something I don't get. I know for instance that the linear polynomial for say f = 91 + 2x + 3y + 8z + Quadratic(x, y, z) + Cubic(x, y, z) ... is 91 + 2x + 3y + 8z if the base point is (0, 0, 0). This is pretty clear. What I don't get is why when you take the base point to be say (1...

For multivariable limits, the way my math books has taught me to prove they exist is to use the epsilon delta argument (for every epsilon > 0, there is a delta >0 ...). I have heard that for most cases you will almost never have to use this argument. Is this true? I know you can use the squeeze...

In textbooks these polynomials are not normally presented as an infinite series (the single variables are). What is the reason for this and are they equally allowed to be in infinite series form hence infinite order just like the single variable Taylor Polynomials? Or are there more issues about...