Multi-variable Definition and 53 Threads

I know that for 1 variable, one can write ##e^{f(x)} = \sum_{n = 0}^{\infty}\frac{(f(x))^n}{n!}##. In the case of 2-variables ##f(x,y)##, I assume we cannot write ##e^{f(x,y)} = \sum_{n = 0}^{\infty}\frac{(f(x,y))^n}{n!}## right (because of how the Taylor series is defined for multiple...

I'm reading Ordinary Differential Equations by Andersson and Böiers, although this is more related to multivariable calculus. There is a Lemma regarding Lipschitz continuity which I have a question about. Below ##\pmb{f}:\mathbf{R}^{n+1}\to \mathbf{R}^n ## is a vector-valued function defined by...

I did a change of variable $$\theta(r,z) = T(r,z)-T_{\infty}$$ which resulted in $$\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial \theta}{\partial r})+\frac{\partial^2 \theta}{\partial z^2}=0$$ $$\left.-k\frac{\partial \theta}{\partial r}\right\rvert_{r=R}=h\theta$$...

Because the limit of the integral is multi-variable, which is not explained at the ML Boas's example, I tried to start from the basic. First, I use: $$\frac {dF}{dx}=f(x) \Rightarrow \int_a^b f(t) dt = F(b) - F(a)$$. In my case now: $$\int_{u(x)}^{v(x,y)} f(t) dt = F(v(x,y)) - F(u(x))$$ So...

Firstly, the matrix notation of the series is, \begin{align*} f\left(x, y, z\right) &= f\left(a, b, c\right) + \left(\mathbf{x} - \mathbf{a}\right)^T \frac{\partial f\left(a, b, c\right)}{\partial \mathbf{x}} + \frac{1}{2}\left(\mathbf{x} - \mathbf{a}\right)^T \frac{\partial^2 f\left(a, b...

Hey, I've got this problem that I've been trying to crack for a while. I can't find any info for multi-variable expected values in my textbook, and I couldn't find a lot of stuff that made sense to me online. Here's the problem. Find $E(C)$ Find $Var(C)$ I tried to get the limits from the...

How can I calculate ∂/∂t(∫01 f(x,t,H(x-t)*a)dt), where a is a constant, H(x) is the Heaviside step function, and f is I know it must have something to do with distributions and the derivative of the Heaviside function which is ∂/∂t(H(t))=δ(x)... but I don't understand how can I work with the...

1)An airplane heads due east at 300 mph through a tailwind whose velocity is given by $w=\langle20,20\rangle$ How fast is the tailwind blowing? In what direction? How fast is the plane flying? In what direction?Answer: The tailwind is blowing at 28.2843 mph approx.in $45^{\circ}$ direction...

Homework Statement Find the volume of the solid between the cone ##z = \sqrt{x^2 + y^2}## and the paraboloid ##z = 12 - x^2 - y^2##. Homework Equations ##x^2 + y^2 = r^2## The Attempt at a Solution I drew a simple diagram to start off with to visualize the solid formed by the intersection of...

Homework Statement If the Green's function of the electric field in a system is G(x,x')=e^{-i(x-x')^2} I want to calculate the phase of the electric field at x if the source is uniformly distributed at x'=-\infty to x'=\infty Homework EquationsThe Attempt at a Solution Then, the phase of...

Hello, currently I am a high school senior who will be going to college in the fall and since my school ends in may and college starts in mid-August. I am planning on self-studying calculus 3, so I can test out of it and go straight into partial differential equations. The textbooks that the...

hey everyone just started university and the jump i feel is huge from a level and was just wondering if you guys knew of any books that had linear algebra and/or several variable calculus in them but displayed and explained stuff in a clear simple way? or if anyone has any websites that do the...

So, i am currently studying physics in a brazilian university. I am going to have a Calculus 2 course which, in Brazil, covers Ordinary Differential Equations and multi-variable differential calculus. So which challenging and rigourous books would you guys recommend for that? Thanks for the...

Homework Statement f(x,y) = 1/y^2-x find the domain of f. Given c ∈ R \ {0} find (x, y) ∈ R 2 such that f(x, y) = c. Finally determine the range of f. Homework Equations I know that the domain of the function is anywhere that the function is defined. The Attempt at a Solution in the case of...

Suppose you have a parameterized muli-varied function of the from ##F[x(t),y(t),\dot{x}(t),\dot{y}(t)]## and asked to find ##\frac{dF}{dt}##, is this the correct expression according to chain rule? I am confused because of the derivative terms involved. ##\frac{dF}{dt}=\frac{\partial...

Homework Statement Evaluate or show that the limit does not exist: \lim_{(x,y) \to (0,0)}\frac{ 2x^{4} + 5y^{3} }{8x^{2}-9y^{3}} \lim_{(x,y) \to (0,-2)}\frac{ xy+2x }{3x^{2}+(y+2)^{2}} Homework EquationsThe Attempt at a Solution So the first one is indeterminate and cannot be factored...

We know that in the space of functions, its possible to find a complete set so that you can write for an arbitrary function f, ## f(x)=\sum_n a_n \phi_n(x) ## and use the orthonormality relations between ## \phi##s to find the coefficients. But is it possible to find a set of functions ##...

Dear Physics Forum personnel, I am a college sophomore with double majors in mathematics & microbiology and an aspiring analytic number theorist. I will be going to self-study the vector calculus by using Hubbard/Hubbard as a main text and Serge Lang as a supplement to Hubbard; this will help...

Homework Statement 2. The attempt at a solution By chain rule, which simpifies to, After this I am struck.

I am looking for a good and easy access reference on multi-variable calculus of variation with many examples and demonstrations. Although I have many books and references on the calculus of variation, most are focused on single-variable. Any advice will be appreciated.

There is a large chunk of information necessary as a preface to my question, so bare with me for a paragraph or two. I work for a pond treatment company. We have a set number of ponds we treat during a month, some are contracted to be treated once a month, some are treated twice. The question is...

For a function of a single variable, I can check if the function is differentiable by simply taking the limit definition of a derivative and if the limit exists, then the function is differentiable at that point. Differentiability also implies continuity at this level.Now, for a function of...

Homework Statement 1. z^6=(64,0) 2. z^4=(3,4) Homework Equations These are expanded out into Real and Imaginary components (treat them seperate): 1. REAL (EQ 1) - x^6-15x^4y^2+15x^2y^4-y^6=64 IMAG (EQ 2) - 6x^5y-20x^3y^3+6xy^5=0 From here, you basically solve these for all six...

Homework Statement I need to find the extrema of f(x,y) = 3x^{2} + y^{2} given the constraint x^{2} + y^{2} = 1 Homework Equations I'm not sure what goes here. I've been trying to solve it with this: ∇f(x,y) = λ∇g(x,y) The Attempt at a Solution f(x,y) = 3x^{2} + y^{2} g(x,y)...

Multi-Variable / Dimension Fourier Decomposition Say we have f(x, y). We can Fourier decompose it in terms of f1(y, v) and e^{\ x\ v}, f2(x, u) and e^{\ u\ y}, or both variables simultaneously f3(u, v) and e^{\ x\ v\ +\ u\ y}. Similarly for any greater number of variables or dimensions. Now, is...

Homework Statement F = 1/2.a(T-Tc)M^2 + 1/4.bM^4 I need to find dF/dM a,Tc,b are positive constants 2. The attempt at a solution I assume this is to do with partial derivatives etc. So I found: ∂F/∂T = 1/2.aM^2 ∂F/∂M = TaM - TcaM + bM^3 And using a chain rule: dF/dM = ∂F/∂T.dT/dM +...

Where is the function $${f (x, y) = x^2 + x y + y^2}$$ continuous? How do I go about solving such problems?

The question asks to differentiate (2nb^{rx}+n)^{k} However, the problem is that it doesn't specify with respect to which variable should the derivative be taken. When the question asks to differentiate, does it mean that we should take the derivative for each and every variable, one by one?

I am currently enrolled in Calculus AB and am going to take the Calculus BC exam. I taught myself most topics in single variable calc over the summer so I am very confident that I can pass the BC exam. Provided I do, I would have the opportunity the take Calc III (and possibly partial diff...

Homework Statement This is a bonus problem on our homework, and I'm having trouble figuring out how to setup what I need. Homework Equations Here are my best guesses: f_x=\frac{\partial f}{\partial x} f_y=\frac{\partial f}{\partial y} f_{xx}=\frac{\partial}{\partial...

How do I find solutions for an equation like: x(1+1/y)+((y**3)/x)-(x**2)(4/y)=(y/2)-y+(4)(y**4)-4 Another, less complicated that I also am confused about is something like: x**3-x/y=1/y Can one simplify all equations to an x=... form? Can these equations be simplified to x=... or some...

I would like to check my answers... Homework Statement Given nonzero vectors u, v, and w, use dot product and cross product notation to describe the following. A vector orthogonal to u X v and u X w A vector orthogonal to u + v and u - v A vector of length |u| in the direction of v...

Homework Statement In real-number multiplication, if uv1 = uv2 and u ≠ 0, then we can cancel the u and conclude that v1 = v2. Does the same rule hold for the dot product: If u • v1 = u • v2 and u ≠ 0, can you conclude that v1 = v2? Give reasons for your answer. Homework Equations...

I would like to check my work with you all. :smile: Homework Statement Let \vec{u} = 2\vec{i}+\vec{j}, \vec{v} = \vec{i}+\vec{j}, and \vec{w} = \vec{i}-\vec{j}. Find scalars a and b such that \vec{u} = a\vec{v}+ b\vec{w}. Homework Equations Standard Unit Vectors: \vec{i}...

So I'm computing a second order Taylor series expansion on a function that has multiple variables. So far I have this I(x,y,t)=dI/dx(change in x)+dI/dy(change in y)+dI/dt(change in t)+2nd order terms Would it still be a better approximation than just he first order if I included some...

Homework Statement The curve p:R->R^{n} and the vector v \in R^{n}. Assume v and p'(t) are orthogonal for all t. And that p(0) is orthogonal to v . Prove that p(t) and v are orthogonal for all t. Homework Equations Since the previous question in the same main question (ie 2(a) and we...

If I have a large amount of data I can sample, with a several discrete variables, and I need to get an average of some function of that data, but it's too computationally intensive to do exhaustively... I want to do some sampling of the possible outcomes. I guess random sampling (Monte-Carlo...

Homework Statement Evaluate the double integral I=int(int(D)( xydA) where D is the triangular region with vertices (0,0)(5,0)(0,3). The Attempt at a Solution I was wondering if my bounds for x would be 0 to 5 and y to be 0 to 3

Hi, Assume that I have f(u,v,w,h) du dv dw dh and I need only to change three variables (u, v, w) say to other variables called (s, r, t) and keep h as is it is So my question can I write this as f(u,v,w,h) du dv dw dh = G(r,s,t,h) J(r,s,t) dr ds dt dh where J is jacobian...

i need some help with this question Find the limit, if it exists, or show that the limit does not exist lim(x,y)->(0,0) x2sin2y/(x2+2y2) i've tried to x=y x=0 or x=y2 but i still got 0...

OKay, so I self-studied Calculus II when I was in High school using a prep book and I did very well. So I am looking into self-studying Calculus III, what good books are there? Do not suggest textbooks because I didn't use textbooks when I self-studied Calculus II.

Homework Statement \int\int of R ( sin( x^2 + y^2) ) dA where the region 4\leq x^2+y^2 \leq 49 Homework Equations not too sure but i know that dy dx = r d(r) d(theta) The Attempt at a Solution i don't understand how to change into polar coordinates in order to integrate. I'm...

Homework Statement Find the mass of the rectangular box B where B is the box determined by 0 \leq x \leq 1, 0 \leq y \leq 2, and 0 \leq z \leq 1, and with density function \rho ( x, y, z ) = z e^{x+y}. Homework Equations "u" substitution The Attempt at a Solution I believe I've...

Homework Statement Using the definition of a limit, prove that lim(x, y) --> (0,0) (x^2*y^2) / (x^2 + 2y^2) = 0 Homework Equations Now, I know that the limit of f(x, y) as (x, y) approaches (a, b) is L such that lim (x, y) --> (a, b) f(x, y) = L. Also, for every number epsilon > 0, there is...

Homework Statement At the end of a much longer problem, I'm asked to find a second height that will satisfy a formula found for a height in the first part of the problem where:Homework Equations h1(h2-h1)=h3(h2-h3)The Attempt at a Solution I know the answer I should get: h3=h2-h1 But I cannot...

I am looking for a good book to go over the basics of Calculus again, currently I am looking at my schools library and have found this one called: Essential Calculus: Early Transcendentals by, James Stewart. My goal is to over the year probably finish this book or if possible even before...

I have a simple question, let's say I have a function f = f(h(a,b), c, d). Can I express this as f = g(a, b, u(c,d))? Are the two expressions equivalent or is one different/more general than the other?

My problem is this. We have three javelin throwers A,B and C. It is known that A defeats B with probability 60%, B defeats C with probability again 60% and A has the better of C with 70%. What are the a-priori probabilities of A, B, C winning the 3-way contest ? If in general we have N...

Let f(x+iy)=u(x,y)+iv(x,y). Suppose we know |f|^2=u^2+v^2 is a constant function. Then we are allowed to say that (u^2+v^2)_x=(u^2+v^2)_y=0. But are we allowed to differentiate u by x and v by y? IE, are we allowed to make the following statement: (u^2)_x+(v^2)_y=0 I'm guessing 'no', but...

Can someone explain to me how to do this: Determine whether the line and plane are perpendicular or parallel or neither x = -1+2t y = 4+t z = 1-t 4x+2y-2z-7=0 My attempt: 2/4 = 1/2 = -1/-2 Since the ratios are the same, does it mean it is parallel? Also when is it perpendicular?