Multivariable calculus Definition and 277 Threads

Homework Statement the base of an aquarium with given volume V is made of slate and the sides are made of glass. if slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials. Homework Equations The Attempt...

Homework Statement What is the maximum possible volume of a rectangular box inscribed in a hemisphere of radius R? Assume that one face of the box lies in the planar base of the hemisphere. NOTE: For this problem, we're not allowed to use Lagrange multipliers, since we technically haven't...

Homework Statement Define f:R2→R3 by f(x,y,z)=(xy+z) ...(x2-yz) let p = (1,1,1)T and h=(δ,ε,θ) a)what are n and m? evaluate f(p) and f(p+h) b)Calculate the Jacobian Matrix Df(x,y,z) and evaluate Df(p) c) Calculate the error e(h) in the first order approximation to f(p+h) d) show...

Homework Statement Hi, Find the line L of the intersection of the two planes x+y+z=1 z-2y+3z=1 What I did was use Gaussian reduction on the augmented matrix. It was easy [x y z] = [3/2 -1/2 0] + z[-5/2 3/2 1] or in equation form or whatever it's called x = 3/2 -5/2 z y = -1/2...

I think i found the solution to my problem but i was hoping to have someone check to make sure i did not make a mistake. \xi = x - ct...... (1) u(t,x) = v(t,\xi)......(2) Taking the derivative d[u(t,x) = v(t,\xi)] \frac{\partial u}{\partial t}dt + \frac{\partial u}{\partial x}dx =...

Homework Statement See attachment Homework Equations The Attempt at a Solution I'm not sure how to determine which points are co linear or which point lies in between the two. My book doesn't discuss how to do this at all. but PQ→ = <1,-1,2>, PR→ = <3,-3,6> I believe I found...

Homework Statement It would help if you guys had access to Maple Anyways here is the problem: http://poibella.org/calc3f11/wp-content/uploads/2011/09/lab_3_vector_calc_F_11.pdf It is on question 12 asking for flux and the questions after that. Homework Equations In the link The...

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...

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?

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 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...

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...

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 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 A 20-inch piece of wire is to be cut into three pieces. From one piece is made a square and from another is made a rectangle with length equal to twice its width. From the third is made an equilateral triangle. How should the wire be cut so that the sum of the three areas is...

Is this a bad idea? I want to get my intro courses out of the way, but don't want to sacrifice a full understanding.

Homework Statement Find the local maximum and minimum values and saddle point(s) of the function. f(x,y) = 1 + 2xy - x^2 - y^2 Homework Equations The Second Derivative Test: let D = D(a,b) = fxx(a,b)*fyy(a,b) - [fxy(a,b)]^2 if D > 0 and fxx(a,b) > 0, then f(a,b) is a local minimum...

Homework Statement Let x=x^2ysin(u)tan(v), where x(u,v) and y(u,v) are smooth functions that, when evaluated at u=1 and v=-3 satisfy x=2.112, y=4.797, \partialx/\partialu = -3.491, \partialx/\partialv = -2.230 , \partialy/\partialu = 1.787 , \partialy/\partialv = 1.554. Then the...

Homework Statement http://img195.imageshack.us/img195/8196/mathqshwk4.jpg Find https://webwork.elearning.ubc.ca/webwork2_files/tmp/equations/f3/d5a53499bbf12cafa144440e3095781.png and [PLAIN]https://webwork.elearning.ubc.ca/webwork2_files/tmp/equations/b1/ce11cb4b480b6159791b1605b6f2681.png...

Hello!I am a student in class 10 at the moment and am eager to do 11 physics but a devilish problem exists...calculus!Could anyone of you suggest me which topics to complete in maths before approaching calculus.Also suggest a few good books to experience calculus.pls!

Homework Statement Find parametric equations for the tangent line at the point (cos (-5*pi/6), sin (-5*pi/6), -5*pi/6) on the curve x(t) = cos t y(t) = sin t z(t) = t (Your line should be parametrized so that it passes through the given point at t=0). Im not really understanding the question...

Homework Statement Suppose u=(-2,-10) and v=(-2,-2) are two vectors that form the sides of a parallelogram. Then the lengths of the two diagonals of the parallelogram are... Homework Equations The Attempt at a Solution I tried using pythagrean theorem and some trig to find the...

Homework Statement Find the equation of the sphere centered at (-7,6,7) with radius 2. Normalize your equations so that the coefficient of x^2 is 1. Homework Equations (x-xo)^2 + (y-yo)^2 + (z-zo)^2=r The Attempt at a Solution (x-(-7))^2 + (y-6)^2 + (z-7)^2 = 2 it saids to...

Homework Statement Find the scalar and vector projection of the vector b=(3,5,3) onto the vector a=(0,1,-5) . Homework Equations The Attempt at a Solution What I've tried is multiplying all the i's and j's and k's together and adding up everything because you get a scalar...

Well, I have a week and a half until my first classes in my first year of college. My math class is multivariable calculus. However, while I aced Calculus in high school, I'm worried that college-level may be of a different caliber. So, I wanted to ask what should I know walking into that...

Hello. I was wondering if I should self study multivariable calculus or introduction to proofs? I am an entering high school senior (contrary to what my username might suggest), and I just took a Calc 2 class last spring. I can only do one or the other, and I don't know which one would be...

Homework Statement We say that a differentiable function f : \mathbb{R}^n \rightarrow \mathbb{R} is homogenous of degree p if, for every \mathbf{x} \in \mathbb{R}^n and every a>0, f(a\mathbf{x}) = a^pf(\mathbf{x}). Show that, if f is homogenous, then \mathbf{x} \cdot \nabla f(\mathbf{x}) = p...

Homework Statement I don't need to state the whole problem; it's the definitions at the beginning that are giving me trouble. Homework Equations So it says, Definition: A function f(x,y) is continuous at a point (x0,y0) if f(x,y) is defined at (x0,y0), and if lim(x,y)-->(x0,y0)...

Homework Statement Prove the following, using the meaning of a limit: Homework Equations epsilon > 0, delta > 0 0 < sqrt(x^2 + y^2) < delta | f(x) - 0 | < epsilon (1) The Attempt at a Solution So, I know that I have to elaborate on the inequality in (1), further. However, I'm...

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...

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...

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...

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...

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 secondary...

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 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...

The problem: Find the value of dz/dx at the point (1,1,1) if the equation xy+z3x-2yz=0 defines z as a function of the two independent variables x and y and the partial derivative exists. I don't know how to approach the z3x part. I thought you would use the product rule and get 3(dz/dx)2x +...

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...

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?

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...