Multivariable Definition and 536 Threads

1. I know both dne but how can i prove it? I am not getting any solid answers? help please! (x,y) to (0,0) 1. ((x^2)y+x(y^2))/((x^2)-(y^2)) 2. (x+y)/((x^2)+y+(y^2)) 2. 1. Simplified down to xy/(x-y) 2. Simplified down to x/(x^2+y^2+y) + 1/((y+1)+x^2)

Problem: If c is in Vn, show that the function f given by f(x) = c.x (c dot x, where both c and x are vectors) is continuous on ℝn. How do I go about proving this? I'm not sure if c is supposed to be a constant or a constant vector, but since it is bolded in the book I am assuming it is a...

Homework Statement Find the length of the curve traced by the given vector function on the indicated interval. r(t)=<t, tcost, tsint> ; 0<t<pi Homework Equations s= ∫||r'(t)||dt The Attempt at a Solution r'(t) = <1, -tsint + cost, tcost + sint> s= ∫||r'(t)||dt ||r'(t)|| =...

Hi, please refer to the attached image. I am having trouble when doing Exercise 2 Here is what I did: $ \int_{-2}^{2}(f(x)\sin(\frac{m\pi x}{2}))dx = \sin(\frac{m\pi x}{2})a_{0} + \int_{-2}^{2} \sum\limits_{n=1}^\infty (a_{n}(\cos(\frac{n\pi x}{2})\sin(\frac{m\pi x}{2})+b_{n}\sin(\frac{n\pi...

Hi all, I am a university student taking Calculus II at the moment. The course sometimes use physics examples, however I do not have any physics background. So I am reaching out to you guys to help me through this math question. Homework Statement A baseball is hit from 4ft above home plate...

Hello MHB, I am working with a limit problem that I get that it does not exist but W|A says it does exist and it is equal to zero... \lim_{(x,y)->(0,0)} \frac{xy^4}{x^2+x^8} well I change to polar and get after simplify \lim_{r->0}\frac{r^3\cos(\theta)sin^4(\theta)}{\cos^2(...

I wanted to know if there is any way of classifying the set of all non-linear multivariable functions. I wish to analyse something over all possible non linear functions with 4 variables. In fact these variables are binary variables. for example f(x,y,u,v)= x.y - u\oplusv

Hello MHB, \lim_{(x,y)->(0,0)} \frac{6x^3y}{2x^4+x^4} I did easy solve that the limit do not exist by (0,t)=0, (t,0)=0, (t,t)=\frac{6}{3} but I wanted Also to solve this by polar cordinate so we got \lim_{r->0}\frac{6\cos(\theta)\sin(\theta)}{2\cos(\theta)+ \sin(\theta)} so My question is what...

Hi! I do not understand the math used in the beginning of this video: In example 1 (4 minutes in the video), why is it wrong to simply solve the problem like this: \vec{V} = [x,-y] \Rightarrow \frac{d\vec{V}}{dt} = [\frac{dx}{dt},-\frac{dy}{dt}] = \vec{a} = [V_x,-V_y], where V_x and V_y are...

1. Homework Statement \text{Let f be a differentiable function from ℝ^2 to ℝ that satisfies:}\ 1) f(x,y)=0\ \text{for all}\ (x,y)\ \text{in the circumference}\ x^2+y^2=2 2) \text{If we consider the function g from ℝ to ℝ^2 given by g(t)=(t+1,e^​t), then the function fog has a relative...

Author: Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum, Daniel E. Flath, Patti Frazer Lock, Thomas W. Tucker, David O. Lomen, David Lovelock, David Mumford, Brad G. Osgood, Douglas Quinney, Karen Rhea, Jeff Tecosky-Feldman Title: Calculus: Single and Multivariable Amazon...

So I'm trying to study for my calc 3 exam, but I have noticed that my book doesn't have many limit questions that use sin/cos/tan, but I know those will likely be on the test. I have tried to google but I can't find many examples that have solutions. Are there any free sites that have problems...

equation of the form ax+bx+cx=a0+b0+c0 ax+bx+cx=d what exactly does "d" represent? is it the distance of a point from the plane? or is it the shifting of the plane along the normal vector of the plane? and how does "d" represent that again?

"As (x,y) \rightarrow (0,0), r \rightarrow," is a fact that I am given, in order to solve a problem. I simply want to know if I properly understand why this fact is true. I know that x = rcos \theta and y=rsin \theta. If I were to look at the individual limits, as x \rightarrow 0 and y...

Homework Statement Take a constant p ≥ 1 and f(x, y) a function of two variables with continuous first order partial derivatives. If, f(λx, λy) = (γ^p)f(x,y) for λ ε ℝ, prove that x(∂f/∂x) + y(∂f/∂y) = pf Homework Equations x(∂f/∂x) + y(∂f/∂y) = pf f(λx, λy) = (λ^p)f(x,y)The Attempt at a...

Hello everybody! My question has to do with the parameterization of 3D surfaces from 3 variables to 2. Specifically, I'm trying to figure out an aspect of the cross product of the directional derivatives of the parameterization to solve flux integrals. Trying to convert: ∫∫SF dS = ∫∫DF(r(u,v))...

Hello, We just started to learn about functions of several variables in my Calculus class and my question is simple: Are conic sections, like ellipses, multivariable functions or is y still dependant on x? Are ellipses just single variable functions slightly rearranged? Thanks in advance...

Hello folks! Just a concise introduction of myself before I get to the task at hand: I'm new to these forums although I have been surfing them frequently for the past 5 years! I am not a math major and quite frankly, my skills in the subject are limited. Be that as it may, my fascination for...

I studied from Multivariable Calculus by James Stewart this past year and thought that it would be worth reading another calculus text to fill in the gaps and to keep my skills sharp. While reading Advanced Calculus by David Widder, I came across this problem: (Paraphrased from text) Suppose a...

Homework Statement Let C be the circle defined by (x-2)2+y2 = 1. If this circle is rotated along the y-axis, a torus will form. What is the Cartesian equation for the torus? The Attempt at a Solution The solution manual says you just switch the x in (x-2)2+y2 = 1 with r=√(x2+z2) and...

If the curvature is always constant and >0 for a parametrized curve C, does it automatically mean the curve is a circle?

Path Integrals-- Multivariable Calculus Hi all-- really stuck here, help would be greatly appreciated. :) 1. Evaluate ∫F
ds (over c), where F(x, y, z) = (y, 2x, y) and the path c is defined by the equation c(t) = (t, t^2, t^3); on [0, 1]: 2. Homework Equations L = sqrt(f'(t)^2 +...

Hello, Given is the function f = f(a,b,t), where a=a(b) and b = b(t). Need to express first and second order derivatives. \frac{\partial f}{\partial a} and \frac{\partial f}{\partial b} should be just that, nothing more to it here, correct? But \frac{df}{dt} = \frac{\partial...

Hello MHB, I am working with finding max and min value and I always hesitate when I got absolute value so the one I strugle is to derivate |\sin(x+y)| if we do it respect to x is this correct? f_x(x,y)=\frac{\sin(2x)}{2\sqrt{\sin^2(x+y)}}|\pi\rangle

Homework Statement Examine lim (x,y) -> (0,0) of: \sqrt{x^{2}+1} - \sqrt{y^{2}-1} \frac{\sqrt{x^{2}+1} - \sqrt{y^{2}-1}}{x^{2}+y^{2}} The Attempt at a Solution Tried variable sub: \sqrt{x^{2}+1} = a, \sqrt{y^{2}-1} = b \frac{a - b}{a^{2}-b^{2}} (a -> 1, b -> 1 as x,y ->...

Homework Statement Homework Equations The Attempt at a Solution I am not really sure how to optimize it from here. What should I be taking the derivative of the price function with respect to ??

For a function z=f(x,y), Keeping y constant, using Taylor's f(x+Δx,y)-f(x,y)=...-equation 1 f(x,y+Δy)-f(x,y)=...-equation 2 Then total change of z =equation 1 + equation 2 Is this correct? but in the picture uploaded...

Homework Statement Find the critical points of f. f(x,y)=2+\sqrt{3(x-1)^2+4(y+1)^2}Homework Equations For fx(x,y) I get: f_x(x,y)=0+\frac{1*6(x-1)}{2\sqrt{3(x-1)^2+4(y+1)^2}}=\frac{3(x-1)}{\sqrt{3(x-1)^2+4(y+1)^2}} For fy(x,y) I get...

Calculate biggest and lowest value to function f(x,y)=x^5y^4e^{-3x-3y} In the triangle has vertices in points \left(0,0 \right),\left(6,0 \right) and \left(0,6 \right) Before I start I want to warn that I used google translate in the text 'In the triangle has vertices in points' Progress: I...

I have the following integral: \int_0^{f(x,y)}{f' \sin(y-f')df'} Now suppose that f(x,y) = x*y, my question is how do I write the integral in terms of x and y only? Can I do something like this? Since df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy we can obtain...

Homework Statement So this is arising in my applied math course in solving the wave equation in n dimensions. So we have a function u(\vec{x}+r\vec{z},t) and where x and z are n dimensional vectors and r is a scalar (also, u is a scalar function). Then when we take the partial derivative with...

Why does it write all that stuff about fxy2? Isn't it unnecessary? I mean, isn't the following true? If in a point (a,b): fx=fy=0 and fxxfyy>0, then the partial derivatives must be either both negative or positive, and thus point (a,b) is a local minima or maxima. And if fxxfyy<0 => (a,b) is a...

As a preface, this question is taken from Vector Calculus 4th Edition by Susan Jane Colley, section 2.3 exercises. Homework Statement "Explain why each of the functions given in Exercises 34-36 is differentiable at every point in its domain." 34. xy - 7x^8y^2 cosx 35. \frac{x + y +...

Homework Statement A function f(x,y)=\mathbb{R}^{2}\rightarrow \mathbb{R} is defined by: f(x,y)=5y^{2}-x^{2} (i) Find its gradient vector and evaluate it at the point (x,y)^{T}=(1,1)^{T}. Find the rate of change of the function in the direction (2,1)^{T} at the point (1,1)^{T} (ii) In what...

Hi, I'm trying to learn physics and math on my own (I'm 16, bored with what I'm taught in school, and on vacation right now, so I have some free time). Over the past 4 months or so I've started more seriously learning, and I think I've been doing fairly well. To learn mechanics, I've been using...

Homework Statement f = x^2 + 4xy + y^2 + 6x + 8 Find minimum, maximum, or saddle point Homework Equations A = [f_xx, f_xy; f_yx, f_yy] The Attempt at a Solution Found the determinant to be zero ( i got [2 ,4; 4, 2] ) so i can't use the eigenvalues to determine the extrema. now...

Homework Statement Find the limit if it exists, or show that the limit does not exist. lim (x,y)-> (1,0) (xy-y)/((x-1)^2+y^2) Homework Equations lim (x,y)-> (a,b) f(x,y) 0<((x-a)^2+(y-b)^2)^1/2<\delta abs(f(x,y)-L)<\epsilon The Attempt at a Solution I tried to prove that it does not...

Calculate the average height above the x-axis of a point in the region 0 <= x <= 1 , 0 <= y <= x^2. I don't know what function to use to find the average height.

Homework Statement f(x,y) = 2x+3y Let \epsilon be any positive number. Show that there is a disk with center (1,1) and radius \delta such that whenever P is in that disk, \left| f(P) - 5\right| < \epsilon. Give \delta as a function of \epsilon. Homework Equations \left| 2x+3y - 5\right| <...

Hello, I've been having some trouble getting some notations straight and hence my question. Usually when I see f(x,y) it means to me there is some variable z produced for any combination of x and y in the domain of the function. So, z=x^2+y^2 I imagine as a paraboloid. So z=f(x,y) ... is a...

Homework Statement Consider the parametrization of a torus: \tau(u,v)=((2+cos(v)cos(u),(2+cosv)sinu,sinv) The distance from the origin to the center of the tube of the torus is 2 and he radius of the tube is 1. Let the coordinates on \mathbb R^3 be (x,y,z) . If p = \tau(u,v) then u is...

multivariable limit of (x,y)--->(1,0) of ln(1+y^2/x^2+xy)) Homework Statement limit of (x,y)--->(1,0) of ln(1+y^2/x^2+xy)) Find the limit, if it exists, or show that the limit does not exist. Homework Equations The Attempt at a Solution so i have: lim(x,y)---->(1,0)...

Homework Statement f(x,y) = x^3 - 3x^2 - 6xy + 7y + y^2, x>=0, y>=0 i) Explain why f attains its minimum value on the quadrant. ii) Find the critical points and classify them Homework Equations df/dx = 3x^2 - 6x -6y df/dy = -6x + 7 +2y d^2f/dx^2 = 6x-6 d^2f/dy^2 = 2 d^2f/(dxdy) =...

Homework Statement The body B = f(x; y; z) belongs to R3, x^2+4y^2+3z^2<or equal to 3, x> or equal to 0 y> or equal to 0 z> or equal to 0 The body B has a volume density: M(x,y,z)= (xyz^2)/(1+x^2+4y^2+3z^2) Calculate the total mass and average density of the body. Sorry if...

Homework Statement Given the function: x*y / (4-x²-2y²) if x²+2y² ≠4 0 if x²+2y² = 4 Check if the function is continuous. Homework Equations The Attempt at a Solution I tried using various ways to see if the result of the limit as (x,y)→(2,0) was the same...

Calculate gradient of f f(x,y)=x^3+2y^3 at point P (1,1) and the directional derivative at P in the direction u of the given vector A -> i-j I tried to attempt this but i honestly don't know where to start. I began to take the partial derivatives of f. I got f'=3x^2dx+6y^2dy, however that...

Hi all, I've got a Calculus III Question Homework Statement Find the derivative zs and zt, where z=sin(x)cos(2y)Homework Equations x=s+t y=s-t The Attempt at a Solution I had a go at the solution and this was what I ended up getting for zs, I ended up getting (cosxcos2y)(1)-2sinxsin2y(1)...

Homework Statement f(x,y) = xy(9x^2 + 3y^2 -16) Find the critical points of the function and their nature (local maximum, local minimum or saddle) Homework Equations The Attempt at a Solution I have partially differentiated the equation into: fx = 27yx^2* + 3y^3 -16y fy =...

Homework Statement lim (x,y) -> (0,0) xy/sqrt(x^2+y^2) = 0 The Attempt at a Solution my understanding of my actual goal here is kind of poor given ε>0 there exist ∂>0 s.t. 0 < sqrt(x^2 + y^2) < ∂ then 0<|f(x,y) - L| < ε | xy/sqrt(x^2 + y^2) - 0 | < ε (xy * sqrt(x^2 + y^2)) /...

Homework Statement Let f(x,y,z)=u(t), where t=xyz. Show that f_{xyz} = F(t) and find F(t). The Attempt at a Solution I'm a little confused about the presentation of the variables in this problem. What does F(t) refer to? This isn't a chain rule question, because it's presented before chain...