Multivariable calculus Definition and 269 Threads

Hi, I took Advanced Calculus and we used 'Calculus, a complete course' by Adams. However, I couldn't learn most of the topics actualy. So, I want to learn Multivariable Calculus; double-triple integrals, Stoke's, Green's Theorem etc. What are the good books you know that I can follow easily? Thanks.

Homework Statement Evaluate the integral, \iiint_E z dzdydz Where E is bounded by, y = 0 z = 0 x + y = 2 y^2 + z^2 = 1 in the first octant.Homework Equations Rearranging y^2 + z^2 = 1 it terms of z , z = \sqrt{1-y^2} The Attempt at a Solution From the given equations I...

Homework Statement Here is my assignment, http://imgur.com/1edJ3g5 I figured it would be easier if we know we are both looking at the same thing! I'm looking for help with question 2. I seem to be having trouble with the integration. Homework Equations r=acosθ x^2 + y^2 + z^2 = a^2...

Homework Statement Hello PF! I'm having some trouble on the last part of my assignment, it's question 4 part "c". Here is a picture of the assignment [http://imgur.com/1edJ3g5] ! I'll post this instead of writing it out so we know that we're all looking at the same thing!Homework Equations The...

1. Marine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the blood increases most rapidly. Suppose that in a certain case, the concentration of blood at a point P(x; y) on the surface of the...

I am currently reading baby Rudin, but I only know single-variable calculus at the moment, so I think it would be a good idea to learn the multi-variable stuff non-rigorously before I do the analysis in Rudin (chapters 9-11). So I was thinking of either getting one of the many 'Mathematical...

Determine whether the points lie in a straight line: 1) A (2,4,2), B (3,7,-2), and C (1,3,3) 2) D (0,-5,5), E (1,-2,4), and F (3,4,2) I'm not sure what method I need to use to show that they are or are not in a straight line. I know that the three points in a are not in a line but those in...

I'm having trouble calculating the dS vector. I know there are multiple ways to find dS but can anyone explain them to me. Or redirect me to a site that can help me with this. I've looked in my book and I've found some info on it but I want additional sources that could maybe explain them a...

Homework Statement Express the iterated integral ∫[0,1]∫[0,1-y^2]∫[0,y] f(x,y,z)dzdxdy a. as a triple integral (i.e., describe the region of integration); b. as an iterated integral in the order z, y, x; c. as an iterated integral in the order y, z, x: The Attempt at a Solution so...

1. Find the volume of the region above the triangle in the xy-plane with vertices (0,0) (1,0) (0,1) and below the surface z =f(x,y)=6xy(1-x-y) My attempt is attached

Homework Statement See in picture Homework Equations What is the final answer? The Attempt at a Solution I know dw/du = df/dx * dx/du + df/dy * dy/du & that dx/du = -8sinu & dy/du = -4sinvsinu Stumped on how to get df/dx and df/dy

Homework Statement Homework Equations The Attempt at a Solution Umm can somebody explain to me what just happened. None of that makes any sense to me what so ever.

Hello guys, I would like to ask some questions regarding my coursework, which is about 2nd ODE and multivariable calculus. Since we have the one-dimensional wave equation and values for the string stretched between x=0 and L=2: 0≤x≤L, t≥0 The string is fixed at both ends so we have ...

1. Homework Statement ∫∫S xz dS where S is the boundary region enclosed by the cylinder y2 + z2 = 9 and the planes x = 0 and x + y = 5. 2. Relevant equation∫∫Sf(x,y,z)dS = ∫∫Df(r(u,v)) * |ru χ rv|dA 3. The Attempt at a Solution I think I have broken this up into 3 surfaces. The...

Hi everyone, this is my first post and so I just want to say thank you in advance for any responses to my question. I recently applied for an internship which will take place over the summer of 2014 and it looks like I have a good chance at actually being accepted for the program. However, I...

Hi! My question is: Find maximum and minimum values of the function: f(x,y) = 2x-y+x^2+y^2 when x^2+y^2 ≤ 4 I would like to solve this without using Lagrange method. I get x=-1 and y=1/2 when using partial derivative and set it equql to 0. I can see that the maximum value...

I'm trying to learn multivariable calculus,and I've heard that one of these is fabulous(the authors have the same name):Advanced Calculus: A Differential Forms Approach by Harold Edwards https://www.amazon.com/dp/0817637079/?tag=pfamazon01-20 or Advanced Calculus of Several Variables by C. H...

Homework Statement Show that any function of the form ##z = f(x + at) + g(x - at)## is a solution to the wave equation ##\frac {\partial^2 z} {\partial t^2} = a^2 \frac {\partial^2 z} {\partial x^2}## [Hint: Let u = x + at, v = x - at] 2. The attempt at a solution My problem with this is...

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...

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...

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?

if space curve C=<f(t),g(t),h(t)>, and v=\frac{dC}{dt}=<\frac{df(t)}{dt},\frac{dg(t)}{dt},\frac{dh(t)}{dt}> Why is curvature defined this way? κ\equiv\frac{d\widehat{T}}{dS} \hat{T}=unit tangent vector S=arc length to elaborate, for a space curve, i understand what \frac{dT}{dt} is, but what...

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))...

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 +...

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 Find the arc length of the curve (t) = (1; 3t2; t3) over the interval 0  t  1. Homework Equations L=sqrt(f'(t)^2+g'(t)^2+...+n'(t)^2) (integrated from a to b) int(udv)=uv-int(vdu) The Attempt at a Solution Seems like it should be fairly straightforward-- the...

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 Let T = f(x, y, z), where dx/dt = 4, dy/dt = 4 and dz/dt = -3 Calculate dT/dt if dT/dx = 4, dT/dy = 7 and dT/dz = 9 Homework Equations dT/dt = ∂T/∂x (dx/dt) + ∂T/∂y (dy/dt) + ∂T/∂z (dz/dt) The Attempt at a Solution I tried to get an explicit formula for T. I...

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 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...

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. I borrowed many multivariable Calculus book so that I can choose one for the next semester. The one I liked most is Multivariable Calculus by Ron Larson. It is full of graphics and colours, somthing that is essential to understand functions of two variables. The only thing is that it does...

I just got to a point in multivariable calculus where I realize I can solve problems in assignments and tests but have no actual idea of what I'm doing. So I started thinking about stuff and came up with a few questions: 1. Is picturing the derivative as the slope of the tangent line to a...

My Calc III teacher this fall gives take home exams because he insists that there isn't enough time to cover all the material in Calc III, so he needs every class hour to teach, thus he gives take-home tests. Knowing people who took his calc III class last fall, and him curving an A to be 70%, I...

I have to do a calculus project by researching somewhere it's applied and writing about it and explaining it thoroughly. I was thinking multivariable calculus / vector calculus was used in satellite orbit math, but I could not find enough on this. Any ideas on where else it's applied that...

Homework Statement Given f(x,y) = xy(x+2y-6), let D be the region in the plane between the hyperbola xy = 4 (let this be g) and the line x+2y-6 = 0 (let this be h). Find the maximum and minimum values of f(x,y) on D. The Attempt at a Solution I first found the critical points for f(x,y)...

Homework Statement Here is the question with the solution: http://dl.dropbox.com/u/64325990/MATH%20253/midterm%202.PNG I don't understand how to do a. How did they know to use x=0 and y=0 for Fx(0,0) and Fy(0,0)? I understand that this is implicit differentiation using Chain Rule where...

Homework Statement Here is the problem with the solution: http://dl.dropbox.com/u/64325990/MATH%20253/Capture.PNG What I don't understand is circled in red, how did they combine dxdy with dydx? Is it with Clairaut's theorem? If it is can someone explain how it works in this case because...

Homework Statement For example in: lim (x,y) -> (0,0) [(xy^2)/(x^2+y^4)] This limit does not exist (according to textbook), but if you use squeeze theorem since y^2<(x^2+y^4) y^2/(x^2+y^4) <= 1 and therefore 0 <= (xy^2)/(x^2+y^4) <= x as x--> 0 so lim (x,y) -> (0,0)...

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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 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...