Multivariable Definition and 524 Threads

So I've started reading Landau's classical mechanics text and I'm at the part where L(v'2) = L(v2 + 2v⋅ε +ε2) and he says to expand it out in powers of ε to get L(v'2) = L(v2) + ∂L/∂(v2) * 2v⋅ε where v and ε are vectors. Can someone explain how this result is obtained in more detail?

Hi. I am a bit confused on the definition of multivariable functions. Say you have ##f(x) = x^2 + x## and ##g(x,a) = x^2 + a## where ##a=x##. Is ##g(x,a)## then a mathematically legal multivariable function? Because if you take ## \frac{\partial f(x)}{\partial x}=2x +1## you'll get a different...

Given: $$y=b-ax^2, \ a>0, \ b>0, \ y \geq 0$$ Find the values of a and b that maximize the area, subject to the constraint that the length of the curve above the x-axis is 10. We integrate over $$\left [- \sqrt{\frac{b}{a}}, \sqrt{\frac{b}{a}} \right ] $$, which gives the area as...

Hi PF! Can anyone help me with showing the following: $$\frac{\partial}{\partial x} \int_{f(x)}^{g(x)} L(x,y)dy = \int_{f(x)}^{g(x)} \frac{\partial L}{\partial x} dy + \frac{\partial g}{\partial x} L(g,y) - \frac{\partial f}{\partial x} L(f,y)$$ I understand this as the fundamental theorem if...

Homework Statement find lim as x,y approach 0 of (10sin(x^2 + y^2)) / (x^2 + y^2) Homework EquationsThe Attempt at a Solution direct substitution yields indeterminate form and so does multiplying by the conjugate. what other methods are there to use?

I am quite new to the topic of multivariable calculus. I came across the concept of "gradient" (∇), and although the treatment was somewhat slapdash, I think I got the hang of it. Consider the following case: ##z = f(x,y,t)## ##∇z = \frac{∂z}{∂t} + \frac{∂z}{∂y} + \frac{∂z}{∂x}## If we're...

1. The problem I am trying to prove the following relation in cartesian coordinates. We were given a hint to use integration by parts, as well as the fact that we know $d \vec r = dx\,dy\,dz$ (volume integral). $$\int f(\vec r)\ \nabla \cdot \vec A(\vec r) \, d \vec r = -\int \vec A(\vec...

Hi everyone. A friend of mine asked for help evaluating this multivariable limit. $\displaystyle \begin{align*} \lim_{(x,y) \to (0,0)} \frac{x\,y^8}{x^3 + y^{12}} \end{align*}$ We got the answer of 0 by converting to polars. $\displaystyle \begin{align*} \lim_{(x,y) \to (0,0)}...

Homework Statement Let R be the solid region that is bounded by two spheres x^2 + y^2 + z^2=1 and x^2 + y^2 + z^2=2. Determine the moment of inertia of R around the x-axis if the mass density per unit volume of R is u=sqrt(x^2 + y^2 + z^2). Homework Equations Moment of Inertia around the...

Homework Statement Let the path C traverse part of the circle or radius 3 at the origin, in a clockwise direction, from (0,-3) to (3,0). Calculate the total mass of a wire in shape C, if the mass density of the wire is u=x^2+4y Homework Equations mass of plate equation= double integral u(x,y)...

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 Apply the definition of the limit to show that \begin{align*} f(x,y) = \frac{x^2\,y\,\left( y - 1 \right) ^2 }{x^2 + \left( y-1 \right) ^2 } = 0\end{align*} I know I'm required to use the epsilon delta method here, no polar stuff either, just straight at it. Homework...

Could someone walk me through how to maximize this 2-variable function wrt z? http://www.wolframalpha.com/input/?i=z+%3D100%2F%281%2B%28root+%28%28x-2%29%5E2+%2B+%28y-3%29%5E2%29%29%29+-+100%2F%281%2B%28root+%28%28x-2%29%5E2+%2B+%28y-3%29%5E2%29%29%5E2%29 I know the set of solutions will...

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...

hi everyone , i don't understand these steps for Taylor Expansion , it has used for state space equations the equations are the approximations for sin and cos the equation for Taylor series is ( i don't understand at all ) please help me if you can

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...

Homework Statement Find the first-octant point P(x,y,z) on the surface closest to the given fixed point Q (0,0,0). The surface x2y2z=4 Homework Equations is the distance along PQ. EDIT: The Attempt at a Solution I get stuck here every time. I feel like I'm just selling myself...

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...

Homework Statement Consider a spherical cap, for which the surface area and volume is A(a,h) = \pi(a^2 +h^2) V(a,h) = \frac{\pi h}{6}(3a^2 +h^2) What would the aspect ratio dA/dV be? The Attempt at a Solution Clearly we would have dA = 2\pi a da + 2\pi h dh dV = \pi ha da +...

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...

Homework Statement completing the squares; ## x^2+y^2+2xy-2x-2y+43 = 0## The Attempt at a Solution I did it as follows, but i would like to know if there is a different 'nicer' method to complete it; ## x^2 + y^2 + 2xy − 2x − 2y + 43 ## ## = (x + y)^2 − 2x − 2y + 43 ## ## = (x + y)^2 −...

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 The limit as x,y → 0 \frac{y^{2}Sin^{2}x}{x^{4}+y^{4}} Homework Equations The Attempt at a Solution There are pretty straight forward but I have a general question about them. So say in the function above I let y=x and let x approach 0. I get 0/0 -...

I guess my first questions is whether saying that a function is differentiable is the same as saying that its derivative is continuous. i.e. if \lim_{x\rightarrow{}a}f'(x)=f'(a) then the function is differentiable at ##a##. Or is it just a matter of the value ##f'(a)## existing? Now my...

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. The problem statement Fill in the blanks ∫ [0,1] ∫ [2x^2,x+1] f(y) dy dx = ∫ [0,1] ( ) dy + ∫ [1,2] ( ) dy The expressions you obtain for the ( ) should not contain integral signs. The brackets are the bounds of integration, and the open parenthesis are the blanks. The Attempt at a...

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

Been a while since I stopped by here... There's one thing about optimization on non-compact sets that's been bugging me for quite a while and I'd love to hear how you perceive things. Say we are optimizing a partially differentiable (and thus continuous) function $f:\mathbb{R^2} \to...

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 Let ##H## be the parallelogram in ##\mathbb{R}^2## whose vertices are ##(1,1), (3,2), (4,5), (2,4).## Find the affine map ##T## which sense ##(0,0)## to ##(1,1), (1,0)## to ##(3,2), (0,1)## to ##(2,4)##. Show that ##J_T=5## (the Jacobian). Use ##T## to convert the integral...

Homework Statement dy/dx = (1+x)/xy solve y(1) = -4 Homework Equations The Attempt at a Solution What threw me was the solve for if y(1) = -4 I grouped variables and then integrated both sides and solved for y. (1/2)y^2 = ln|x|+x+c y=+- √2ln|x|+x+c I then...

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 ...

Homework Statement Find the shortest distance from the origin to the surface x=yz+10 Homework Equations The Attempt at a Solution So I said that my main function, f(x,y,z) = x^2 + y^2 + z^2 (the function I want to minimize) Then I said that g(x,y,z) is my constraint function...

I am currently taking Calc III as an online course (yes, big mistake). I am at a section where we are evaluating the domains and ranges for functions with both x and y variables in it. As far as finding domains, no problem. However, the textbook doesn't explain how to solve for ranges...

Homework Statement Three masses are disposed in a circular lane and linked with springs (resting lengths Lo). Find the potential energy and, from it, the equilibrium positions. (see image: https://www.dropbox.com/s/evqcspwlj68p5n9/2014-02-05%2016.07.07.jpg ) Homework Equations The...

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...

Homework Statement I'm having a problem with solving the domain of |x^2+y^2|<=|z^2|. Homework Equations The Attempt at a Solution From what I got this should be separated into two expressions: x^2+y^2<=z^2, and x^2+y^2>=-z^2. The later doesn't have a real solution because of the...

Ok the first problem is The output Q of an economic system subject to two inputs, such as labor L and capital K, soften modeled by the Cobb-Douglas production function Q(L;K) = cLaKb, where a; b and c are positive real numbers. When a+b = 1, the case is called constant returns to scale. Suppose...

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 Given that the surface x^{6}y^{5}+y^{4}z^{5}+z^{9}x^{7}+4xyz=7 has the equation z = f(x, y) in a neighborhood of the point (1, 1, 1) with f(x,y) differentiable, find: \displaystyle\frac{\partial^{2} f}{\partial x^{2}}(1,1) = ? Homework Equations The Attempt at a Solution...

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 A function f is defined on the whole of the xy-plane as follows: f(x,y) = 0 if x=0 f(x,y) = 0 if y = 0 f(x,y) = g(x,y)/(x^2 + y^2) otherwise a) g(x,y) = 5x^3sin(y) b) g(x,y) = 6x^3 + y^3 c) g(x,y) = 8xy For each of the following functions g determine if the...

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)|| =...