What is Multivariable: Definition and 536 Discussions

I'm taking multi-var calculus next semester and I found out that we are using this terrible excuse for a book: https://www.amazon.com/dp/0470131586/?tag=pfamazon01-20 Anyway, I have been studying ahead using my huge copy of Stuart, which my university uses for Calc I and II. I asked my...

Homework Statement Hello all, I have been working on a 3rd order taylor series, but the formula I have does not seem to get me the right answer. The formula I was given is for a taylor polynomial about point (a,b) is: P_3=f(a,b) +\left( f_{1}(a,b)x+f_{2}(a,b)y\right)...

Homework Statement If u=x^3 + 3xy + y^3, determine du/dx on the ellipse 2x^2+3y^2=1 2. The attempt at a solution Imagine I just use partial derivative somehow, but I'm not sure what the question is asking by on the ellipse. I have a feeling its just something simple that I'm...

Hi there. In the multivariate case it is proven that if X~Nm(U, S) and Y = a + CX where C is an invertible n x n matrix, and S is the covariance matrix then: Y ~ Nm(a + CU, CSC') I am trying to apply this proof to a similar problem. If we have X1...Xk with Xi ~ Nm(Ui, Si) where i = 1...k...

Homework Statement Find extreme values for f(x, y, z)=x+y+z subject to: x^4+y^4+z^4=c>0 Therefore, since c>0, exclude the origin. Let L(x, y, z, \lambda{)}=x+y+z+\lambda{(}x^4+y^4+z^4-c) and thus L_1(x, y, z, \lambda{)}=1+4x^3\lambda L_2(x, y, z, \lambda{)}=1+4y^3\lambda L_3(x, y...

Suppose I have F(x,y) and y=y(t) and x=x(t) Therefore, Ft = Fx*xt + Fy*yt. Right? Can I write Ftt = (Fxx*xt + Fyy*yt)*xt + Fx*xtt + (Fxx*xt + Fyy*yt)*yt + Fy*ytt ? Basically I'm trying to figure out the second derivative by chain rule.

Homework Statement z^{3}x+z-2y-1=0 xz+y-x^{2}+5=0 Define z as a function of x, find z'. Homework Equations I guess the two equations above... The Attempt at a Solution Well, I just differentiated the first one with respect to x and got: 3z^{2}xz'+z^{3}3+1=0 z' = \frac{-1-z^{3}{3z^{2}x}...

Hi, I've been working on this problem for a few days and I'm pretty stuck. I'm working on how much area is eclipsed by some arbitrary planet by another over a given time. My question is if I have two circles one given by (y-100)^2+x^2 = 100^2 and the other given by (y+100)^2+x^2 = 100^2...

I was wondering how to prove the multivariable chain rule \frac{\mbox{d}z}{\mbox{d}t}=\frac{\partial z}{\partial y}\frac{\mbox{d}y}{\mbox{d}t}+\frac{\partial z}{\partial x}\frac{\mbox{d}x}{\mbox{d}t} where z=z(x(t),y(t)) I don't really need an extremely rigorous proof, but a slightly...

I am looking for a solution to a multivariable recursive formula as follows: Define: Initial Conditions: a(1,0)=-3, a(2,0)=2, a(n,0)=0 for n≠1,2 Recursive Formula: a(n,k)=(4n+1)a(n,k-1)-4a(n-1,k-1) Find a formula in terms of n and k for a(n,k). This is my first post and so I...

Homework Statement Find the global maximum and global minimum of f(x, y)=x^3+y^2 in the half disk x^2+y^2\leq{1}, y\geq{0} This is a rather new topic for me, so I am looking for clarification for problems such as these. The Attempt at a Solution First, I determined the critical points of the...

Homework Statement \ Find the critical points of the function. Then use the Second Derivative Test to classify each critical point (or state that the test fails). f(x, y) = x3 + y4 - 21x - 18y2 The Attempt at a Solution partial x derivative=3x^2-21=0 partial y= 4y^3-36y=0...

I have a scalar function f dependent on a few variables $x_i$, and I would like to change variables, so that y_i = \sum_j {M_{ij} x_j}, where M is an invertible matrix independent of the x_i-s, and compute: \frac{\partial f}{\partial x_i} = \frac{\partial f}{\partial \left( \sum_j...

My school requires Calculus III before DEQ, though I had the department allow me to take the two concurrently. I am wondering what Calc III topics I should be fluent into prepare for Differential Equations. I'm sure this university teaches DEQ with an understood previous knowledge in...

Homework Statement Stumped. Integral: f(x,y) = ∫ (from 1 to xy) of e^(t^2)dt find both fx and fy The Attempt at a Solution I've come up with: fx(x,y) = ∂/∂x ∫ (from 1 to xy) of e^(t^2)dt Not sure where to go... possibly take the integral, the take the partial derivative? I...

Homework Statement This is an example taken from the textbook lesson and there's one part I don't understand: Find the area of the region bounded above by the spiral r = pi/(3θ) and below by the polar axis, between r = 1 and r = 2. SOLUTION: Double integral of r(dθ)(dr) with boundaries...

I am asked to find all local maximum and minimum points for the function, f(x,y) = (1+xy)(x+y) so, naturally, I had to find the critical points. However, in the back of my book (Calculus Early Transcendentals 6E, James Stewart) the only critical points are (1,-1) (-1,1) (both of which are...

I'm working on some voltage/current physics problems and I think I understand the concepts however I believe the algebra is holding me back(not my strong point). Could someone make some suggestions how I could solve for at least one of these variables by using a combination of theses three...

Homework Statement \lim_{\substack{x\rightarrow 2\\y\rightarrow 2}} f(x,y)=\frac{2x^2+2xy+2x-xy^2-y^3-y^2}{2x^3-2x^2y+2x-x^2y^2+xy^3-y^2} Homework Equations N/A The Attempt at a Solution Well, I tried using the line y = 2, and let x approach 2 (as well as making x=2 and let y...

Homework Statement Suppose a mountain is described by the function z = 10x^2 * y − 5x^2 − 4y^2 − x^4 − 2y^4 and that you are standing at the point (1,1,−2). The positive x-axis points east and the positive y- axis points north. If you walk in the northeast direction what angle above the...

Homework Statement Evaluate the following limits or determine that the limit does not exist. b) lim (x,y)–>(0,0) (1+xy)^(1/xy) Homework Equations The Attempt at a Solution I have a funny feeling this limit might exist. (Then again I get that feeling about every indefinite...

Homework Statement Hello, I need help on the following questions, mainly to make sure i did them right. for both, lim (x,y) -> (0,0) 1) http://www4d.wolframalpha.com/Calculate/MSP/MSP3619hd2e84h398acc80000110c0eac25d415ha?MSPStoreType=image/gif&s=57&w=72&h=44 2)...

Homework Statement lim (x,y) --> (0,0) [sin(2x) -2x + y]/[x^3+y] Homework Equations The Attempt at a Solution so I tried approaching from the x-axis and I got y=0 lim x-> 0 [sin(2x) -x + 0]/[x^3 + 0] which gives me 0/0 and from the y axis y=0 lim y ->0 which gives...

Hello, I'm currently trying to get my Ti-89 Titanium to compute the following triple integral; f(x,y,z) = xy(1-z) where 0<x<1, 0<y<1, 0<z<1, and x+y+z<1 int(int(int(f(x,y,z),x,0,1-y-z),y,0,1-z),x,0,1) The problem is that I can't get my Ti89 to substitute in for the limits of...

Homework Statement Find the limit of \lim_{(x,y)\rightarrow (0,0)}\frac{x^2y^2}{x^3+y^3} Homework Equations I'd like to solve this in a rather elementary manner, so preferably only using the squeeze theorem or through proving the limit doesn't exist via multiple path approach. The...

I need to show that limit (|x|^a*|y|^b) / (|x|^c+|y|^d) = 0 (x,y)->(0,0) when a,b>=0; c,d>0; with a/c + b/d > 1 Does anyone have some tips for starting off the proof?

So the function I'm working with is: f(x,y) = 1/sqrt(1-x^2-4y^2) First, they want me to find the Domain and Range, which I found to be: D: x^2 + 4y^2 < 1 R: (0,1] Then they want me to sketch level curves and cross sections, then sketch f(x,y) I'm having trouble with the sketching, I...

Let a; b and c be any three vectors in R3 with a =6 0. Show that b = c if and only if a
dot b = a dot
c and a cross b = a cross c

What is a good, solid reference text for multivariable calculus? Ideally something you'd get by following on from Spivak's Calculus into 2/3d/Vectors etc. I've already done the course, but it's sort of slipping my mind these days - I've mostly been doing chemistry subjects, and they're not...

Homework Statement Partial derivative of R = vt cos x with respect to t Homework Equations NA The Attempt at a Solution Keeping the other variables constant, I get 'v cos x'. (t is 1) I just want to check this, because it's part of a larger problem and I feel my answer is way...

Homework Statement Evaluate by changining to polar coordinates : #19 in this picture http://i52.tinypic.com/2ngbt5z.jpg Homework Equations x = cos θ y = sin θ r^2 = x^2 + y^2 + z^2 ∫∫∫w f(x,y) dxdy = ∫from θ1 to θ2 ∫from r1 to r2 f(cosθ, sinθ) (r dr dθ)The Attempt at a Solution The first...

Homework Statement Evaluate by changining to polar coordinates : #19 in this picture http://i52.tinypic.com/2ngbt5z.jpg Homework Equations x = cos θ y = sin θ z = z r^2 = x^2 + y^2 + z^2 ∫∫∫w f(x,y) dxdy = ∫from θ1 to θ2 ∫from r1 to r2 f(cosθ, sinθ, z) (r* dr dθ)The Attempt at a Solution The...

Homework Statement Using spherical coordinates, set up but DO NOT EVALUATE the triple integral of f(x,y,z) = x(x^2+y^2+z^2)^(-3/2) over the ball x^2 + y^2 + z^2 ≤ 16 where 2 ≤ z. Homework Equations x = ρ sin ϕ cos θ y = ρ sin ϕ sin θ z = ρ cos ϕ ρ^2 = x^2 + y^2 + z^2 ∫∫∫w...

Homework Statement I'm supposed to find (and classify, which I'm really not sure what that means) critical points of f(x,y) = x2y- x2 - 2y2 - 2x4 - 3y4 Homework Equations The Attempt at a Solution I found the partial derivatives, fx(x,y) = 2xy - 2x - 8x3 and fy(x,y) = x2 - 4y -...

Homework Statement Hi. I'm going to provide two homework examples. The first one I got correctly. But the second one, I used the same method as the first question but I got the answer incorrect. I'm really confused. The first question was \lim_{(x,y)\to(0,0)}\frac{(xycosy)}{3x^2+y^2}...

what is a good multi variable calculus text i have just finished reading spivak's calculus thanks

Homework Statement The position on the ground in the xy plane that is hit by the sun given by (x,y)=(3t+tan(phi), -2t+tan(theta)), where t, phi, and theta, are controlled input variables. What is the velocity of the hit point if the input variables are at values (5, pi/4, pi/3) and changing...

Homework Statement dv/dt = (k+v2)/h Homework Equations k is a constant, and v and h are variables where h is independent of v but v is dependent of h (v is a function of h and t). The Attempt at a Solution dv/(k+v^2) = dt/h. The problem I have is with dealing with the right side...

Hello all, I am currently in University and I am retaking a course that I dropped last year due to the fact that there was no textbook and the time I spent in the course was a frantic scramble for reliable information and attempting to understand what the professor was teaching. I have looked...

1. I am working on an assignment in maple for my Calc 3 class. It seems really simple, but I have no experience using Maple. Any help would be very much appreciated. PART 1: 3D – VECTORS IN MAPLE 1. Let u= i + 2j. Define a vector v. Draw the two vectors in a Maple graph. 2. Use Maple to compute...

Homework Statement The function is defined in the xy-plane. Find all critical points and determine their characteristics. Homework Equations f(x, y) = x4 + 4xy + y4 The Attempt at a Solution I tried finding the partial derivatives. f'x = 4x3 - 4y f'y = 4y3 - 4x Setting them to 0...

Homework Statement Let \textbf{F}: \textbf{R}^m \rightarrow \textbf{R}^n and \textbf{G}: \textbf{R}^p \rightarrow \textbf{R}^m Prove that ({\textbf{F} \circ \textbf{G}})'(x) = {\textbf{F}}'(\textbf{G}(\textbf{x})) {\textbf{G}}'(\textbf{x}) Homework Equations Assume the single...

Hi everyone, I am interested in applying to be a teaching assistant in my university's multivariable calculus class or linear algebra class. I took both of these courses already and seem to like them equally well. I will have solid recommendations and grades. Any recommendations on which...

Homework Statement Define f : R2 -> R by f (x, y) = x²y x4+y2 (x, y) ≠ (0, 0) 0 (x, y) = (0, 0). (i)What value does f (x, y) take on the coordinate axes? (ii) Define g : R -> R2 by g(t) = ( t ) ( kt ) k is an arbitrary nonzero constant...

Define f : R2 -> R3 as f (x, y) = ( xy ) ( y+x2) ( 1 ) Let p = (0, 1)T and h = (δ,ε)T (i) Evaluate f (p) and f (p + h) (ii) Calculate the Jacobian matrix Df and evaluate Df (p) (iii) Calculate the first order approximation to f (p + h), namely f (p) + Df (p)h. Show that...

Define f : R2 -> R by f (x, y) = x²y x4+y2 (x, y) ≠ (0, 0) 0 (x, y) = (0, 0). (i)What value does f (x, y) take on the coordinate axes? (ii) Define g : R -> R2 by g(t) = ( t ) ( kt ) k is an arbitrary nonzero...

Homework Statement The function is f(x,y) = xy + x + y. The region R is defined as all the points lying inside x^2 + y^2 < 8. The problem asks us to find all the points in the region R where f may have a minimum or a maximum. Homework Equations Partial derivatives etc The Attempt at...

Homework Statement Find the limit of ((x^3 + (4x^2)y)/(x^2+2y^2)) as (x,y) -> (0,0) Homework Equations The Attempt at a Solution I am guessing the limit is equal to 0, and I know I have to use 0 < sqrt(x^2 + y^2) < delta where |f(x,y) - L| < epsilon I just have no idea what to do next I...

Hi! I'm studying complex analysis and I don't quite understand why the complex derivative was defined the way it is, and how it is related to the definition of the derivative of a real multivariable function.

Homework Statement Evaluate the following limit or give a reason explaining why the limit does not exist. \lim_{(x,y) \to (0,0)}\frac{x-y}{x+y} Homework Equations x = r*\cos\theta y = r*\sin\theta The Attempt at a Solution \lim_{r \to...