What is Multivariable: Definition and 536 Discussions

I realized I placed this in the wrong forum... I will put it in coursework help

Homework Statement Let g(x, y) = f(sin(y), cos(x)). Find the second partial derivative of g with respect to x (g_xx). Homework EquationsThe Attempt at a Solution I attempted to find g_x, but I'm not entirely sure how chain rule applies in this situation. Is this correct? g_x = f_x(sin(y)...

Hi, I am a math major, and currently studying vector calculus. Since I am feeling that I don't really learn it properly, I am going to re-learn it again in the summer. I would like to improve both my theoretical and computational skills. I am also searching for a book that starts from the...

So this is a problem for microeconomics, but should follow under general calculus: The point is x=(x1(p1,p2,u),x2(p1,p2,u)) where u is a constant on the function u(x1,x2). p1 is the price of x1 and p2 for x2. I'm supposed to show that (dx1/dp2)=(dx2/dp1). I've been given the info that for the...

T(t) = Ts+(98.6 – Ts)e-kt rewrite in the form t=g-1(T) In trying to understand how to find the inverse of this but am having a hard time, please advise. Thanks, Kupkake303

Homework Statement Find the limit \lim_{(x,y)\to(2,2)}\frac{x^3-y^3}{x-y} Homework Equations \epsilon - \delta, baby: If the limit L exists, \forall \: \epsilon \: \exists \: \delta: 0 < \sqrt{(x-a)^2+(y-b)^2} < \delta \rightarrow |f(x,y)-L| < \epsilon The Attempt at a Solution By...

Homework Statement f(x,y) = x2-4xy+5y2-8y ; enclosed by the triangular region (0,0), (3,0), and (3,3) Find the absolute maximum and minimum Homework Equations N/A The Attempt at a Solution First I took the partial derivatives with respect to x and y: fx = 2x - 4y fy = -4x + 10y -8 I...

The problem is: The temperature (in degrees Celsius) of a metal plate, located in the xy -plane, at any point (x, y ) is given by the function of two variables T(x, y ) = x sin y + y2 sin x. (a) Find the rate of change in temperature in the direction of the positive x-axis at the point (π, π)...

Hello Forum, I think I am clear on how to find the maxima and minima of function of one or two independent variables like f(x) or f(x,y). What if the function had 5 independent variables, i.e. f(x,y,z,a,b)? What is the best method, even numerically? Should we find the partial derivatives of f...

As a undergrad future scientist, in my free time I like to do some programming to get some practice as my education programme doesn't include it, but it is a useful skill to have. I am programming some type of roguelike game, so I can get away from graphics and focus on procedural generation...

Homework Statement From the transformation from polar to Cartesian coordinates, show that \begin{equation} \frac{\partial}{\partial x} = \cosφ \frac{\partial}{\partial r} - \frac{\sinφ}{r} \frac{\partial}{\partialφ} \end{equation} Homework Equations The transformation from polar to Cartesian...

I was trying to understand a proof in multivariable calculus (about fundamental pde's), and at some point it said that if ##U## is a convex open set of ##\mathbb{R}^2##, then it's second projection is an open set in ##\mathbb{R}##, without any further explanation. Could you explain please ?

Say I have a function of three variables, ##F=F(s_{12},s_{23},s_{13}) = F(s,t,-s-t)##, where ##s_{12}=s,s_{23}=t## and ##s_{13}=u = -s-t##. I want to compute the differential operators $$\frac{\partial}{\partial s}, \frac{\partial}{\partial t}\,\,\text{and}\,\,\frac{\partial}{\partial u}.$$...

Homework Statement Homework Equations lim(x,y)->(a,b)f(x,y) continuous at (a,b) if lim(x,y)->(a,b)f(x,y)=f(a,b) Squeeze theorem if lim a=lim c and lim a<= lim b <= lim c then lim b= lim c The Attempt at a Solution I proved that all the limits exist but somewhat the functions aren't all...

How do you find a normal vector of a function at a point, such as f(x,y)= ax^y+yx^y^x+b at (X_o,Y_o) where a and b are just arbitrary constants, and the function is an arbitrary function. So I guess, what is the general steps you take to find the normal? I thought it had to do with the...

Homework Statement f(x,y) = (xy) / (x2 + y4), when (x, y) ≠ (0,0) 0, when (x,y) = (0,0) Homework Equations Explicitly show that f(x,y) does not satisfy lim h -> 0 [ E(v,h) / ||h|| ] = 0 when v = 0 (h, v, and 0 are all vectors; I'm not sure how to put a hat on them) The Attempt at a Solution I...

Homework Statement Evaluate: ∇ • (r^3 * ȓ ) (del dot (r^3 times vector r)), where r = sqrt(x^2+y^2+z^2) and vector r = (x, y, z) The Attempt at a Solution [/B] So, taking the partial derivative of the x component, I got 2r3/2. Doing the same thing for the y, z components I got a similar...

I'm trying to replicate the model presented in this [paper](http://www.sciencedirect.com/science/article/pii/S1359431103000474), which is basically to model heat and mass transfer along a one-dimensional duct. There are four characteristic equations for this problem : Momentum conservation...

Hi, I'm looking for a good book to study multivariable calculus from that would fit a proof-based undergraduate math course. The emphasis should be on RIGOROUS proofs, theorems and presentation of concepts. I really need a book that would be approachable for self-study with clear and...

Homework Statement A hill is described with the following function: f(x,y) = 3/(1+x2 +y2) Where f(x,y) is the height. Find the points where the hill is steepest! Homework Equations ∇f(x,y) = d/dx(f(x,y))i + d/dy(f(x,y))j The Attempt at a Solution d/dx(f(x,y)) = -6x/(1+x2+y2)2 d/dy(f(x,y)) =...

Let's call our functional $$F[f]=\int dx\:A\left(x,f,f',f''...\right)$$ We know that the variation of F can be written as $$\delta F=\int dx\:\left[\frac{\partial A}{\partial f}\delta f+\frac{\partial A}{\partial f'}\delta f'+...\right]$$ If i wanted to get everything in terms of delta f in...

Homework Statement Given f(x,y)=(y+x)/(y-x) use an ε-∂ proof to show that lim(x,y)→(0,1) f(x,y) exists. Homework Equations |(y+x)/(y-x)-1|=|(2x)/(y-x)| The Attempt at a Solution I know that the limit is 1. I can't figure out how to massage the above any further to get it into the form...

Dear Physics Forum advisers, Could you recommend books that treat the multivariable calculus from a theoretical aspect (and applications too, if possible)? I have been reading Rudin's PMA and Apostol's Mathematical Analysis, but their treatment of vector calculus is very confusing and not...

Homework Statement Let ##M## be the set of all points ##(x,y) \in \mathbb{R}^2## satisfying the equation ##xy^3 + \frac{x^4}{4} + \frac{y^4}{4} = 1 ## Prove that ##M## is a manifold. What is the dimension of ##M##? Homework EquationsThe Attempt at a Solution I think this question it started...

I am trying to finish the last chapter of Spivak's Calculus on Manifolds book. I am stuck in trying to understand something that seems like it's supposed to be trivial but I can't figure it out. Suppose M is a manifold and \omega is a p-form on M. If f: W \rightarrow \mathbb{R}^n is a...

I'm looking for a well written textbook for multivariable calculus textbook. For single variable I worked with Stewart's calculus mostly and I found it very good. However I took a glance at Stewart's multivariable calculus textbook and I didn't enjoy it. The explanations weren't all that good (I...

Homework Statement Let P be the tangent to the graph of g(x,y) = 8-2x^2-3y^2 at the point (1, 2, -6). Let f(x,y) = 4-x^2-y^2. Find the point on the graph of f which has tangent plane parallel to P.Homework Equations g(x,y) = 8-2x^2-3y^2 at (1, 2, -6) f(x,y) = 4-x^2-y^2 The Attempt at a...

Find the volume of the solid that is enclosed by the cone $z=\sqrt{x^2+y^2}$ and the sphere $x^2+y^2+z^2=2$. The integral is not difficult to set up using cylindrical coordinates, but I'm trying to get a better visualization of the volume I'm actually integrating. I can't seem to get Wolfram...

I have two problems I need help with 1. Homework Statement https://ccle.ucla.edu/mod/resource/view.php?id=801511 https://ccle.ucla.edu/mod/resource/view.php?id=778704 2,3. Answers and work are givenIn the surface integral problem, I do not understand how it sets it up for Method 1. (How does...

Hi, I'm currently having a lot of trouble with this probability problem. For example: Suppose there are 5 balls in a bag with number 1,2,3,4,5. I pick a ball at random 20 times (with replacement). Lets say the probability of each ball being picked is: P(1) = 0.5 P(2) = 0.15 P(3) = 0.1 P(4) = 0.2...

More than just a few problems that happen to pop up in the textbook, I mean.

So for a function of a single variable How can this be extended to the integration of the total differential of a multivariable function over a region (specifically one of two variables)? That is, how do you integrate Say over the circular region ,

Homework Statement Find the minimum value of f(x,y)=e^{x+y}-2 within x≥0 and y≥0. Homework Equations D=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2 Answer is -1 The Attempt at a Solution So for all partial derivatives I got e^{x+y} (and mixed), but when I calculate the discriminate (subbing in...

So i did fairly well in both calc I and II, both being proof based (for math majors, and I got an A in both). I am working a job that allows me to read at work this summer, and would like to teach myself some multivariable/vector calc. I am a physics major, but my school requires me to take the...

Hi guys, just having some confusions on the Delta-Epsilon proofs for multivariable limit functions. here is my question: Apply Delta-Epsilon proof for the Lim (x,y) --> (0,0) of (y^3 + 5x^2y)/(y^2 + 3y^2) to show the limit exists. The part that has me confused is the y to the power of 3, where...

Dear PF personnel, I am a college sophomore with double majors in mathematics and microbiology. I wrote this email to seek your advice on selecting a theoretical, proof-based textbook on the multivariable calculus. I will be taking a multivariable calculus on this Summer but it unfortunately...

Homework Statement Specify the domain and range of f(x, y) = arccos(y − x2). Indicate whether the domain is (i) open or closed, and (ii) bounded or unbounded. Give a clear reason in each case.Homework EquationsThe Attempt at a Solution y-x2 >= -1 y >= x2 -1 y-x2 <= 1 y <= x2 +1 I sketched it...

Homework Statement So the test is to take the determinant (D) of the Hessian matrix of your multivar function. Then if D>0 & fxx>0 it's a min point, if D>0 & fxx<0 it's a max point. For D<0 it's a saddle point, and D=0 gives no information. My question is, what happens if fxx=0? Is that...

1. Problem Define a function: for t>=0, f(x,t) = { x for 0 <= x <= sqrt(t), -x + 2sqrt(t) for sqrt(t) <= x <= 2sqrt(t), 0 elsewhere} for t<0 f(x,t) = - f(x,|t|) Show that f is continuous in R^2. Show that f_t (x, 0) = 0 for all x. Then define g(t) = integral[f(x,t)dx] from -1 to 1. Show...

In my multivariable calculus class, we briefly went over Taylor polynomial approximations for functions of two variables. My professor said that the second degree terms include any of the following: $$x^2, y^2, xy$$ What surprised me was the fact that xy was listed as a nonlinear term. In...

I do not know multivariable calculus. I have studied out of Apostol Vol.1. I do not want to learn the material from Apostol Vol. II. Therefore I want to know If it would be worthwhile to go through Hubbard and Hubbard's 'Vector Calculus, Linear Algebra, and Differential Forms' after going...

How do I justify that lim_{(x,y)\to (0,0)} \cos{\frac{x}{\sqrt{y}}} = 1? If I approach from the y axis, it would become lim_{y\to 0} \cos{\frac{0}{\sqrt{y}}} = 1 , but if I approach from the x axis, it would become lim_{x\to 0} \cos{\frac{x}{\sqrt{0}}} = D.N.E, no? (does not exist) Wolfram...

Hi guys, I have a question about the book "Calculus: An Intuitive and Physical Approach" by Morris Kline. I was wondering if it includes a sufficient coverage of multivariable calculus as well as single variable? I am about to take a course in calculus of several variables and I am a bit...

Homework Statement My teacher introduced the third dimension (## R^3 ##) and higher dimensions to my class using vectors. Later on, my teacher introduced functions of two or more variables and now there's no mention of vectors. I am confused as to how vectors (i + j + k) and functions of two...

Homework Statement https://fbcdn-sphotos-c-a.akamaihd.net/hphotos-ak-xap1/v/t1.0-9/10923273_407123639463753_2874228726948727052_n.jpg?oh=27c882da16071e65bbb420147333ec38&oe=558413E4&__gda__=1434978872_d03c8531060688181560956b68c96650 Is f continuous at (0,0)? What is the "maximum" region D...

Hello, I am taking integral calculus at the moment at my college, and I was just wondering what is the difference between integral calculus and multivariable calculus?

Homework Statement Lim (x,y) --> (pi, 0) of (cos(x-y))/(cos(x+y)) Homework Equations The answer is 1 The Attempt at a Solution My answer is this: The function is continuous at the point in question, so we only need to plug in the values which result to be 1. My question here: I know this...

Homework Statement Lim (x,y,)--> (1, 3) of (x^2-1)/(xy-y) Homework Equations I know that the answer is 2/3 according to wolfram alpha multivariable limit calculator. The Attempt at a Solution So this is my first time doing multivariable limits, I've studied the following: 1) Direct...

I am in Calculus BC in high school right now and I am really enjoying it and have finished all of the material for the year and I have heard different things about what class comes after Calculus BC (I believe BC is equivalent to Calc 1 & 2 in college). I have heard the next class, aka Calc 3...

Homework Statement Here it is: Let Ω be a convex region in R2 and let L be a line segment of length ι that connects points on the boundary of Ω. As we move one end of L around the boundary, the other end will also move about this boundary, and the midpoint of L will trace out a curve within Ω...