Partial derivatives Definition and 417 Threads

In statistical mechanics we express partial derivatives of functions, keeping some variables fixed. But these variables are functions of the other variables (which are not fixed). I'm just confused by this, what is the convention for taking these derivatives? For example, if we have S as a...

How does one take the partial derivatives of a function that is defined implicitly? For example, the function, x^2 / 4 + y^2 + z^2 = 3.

I need to use partial derivatives to prove that u(x,t)=f(x+at)+g(x-at) is a solution to: u_{tt}=a^{2}u_{xx} I'm stuck on how I'm supposed to approach the problem. I'm lost as to what order I should do the derivations in. I tried making a tree diagram, and I came out like this. The arrow...

sorry folks i don't even have an idea to this question`s solution so i hope u people may like to help me. i`m stuck to it since last week nd i hope its from partial derivative... please suggest me a book or a hint or the solution. Let a long circular cylinder of unit radius be placed in a large...

Homework Statement Show that the expression A, T(dP/dT)|V - P is equal to expression B, T^2 * [d(P/T)/dT]|V Also, show that expression C, -[d(P/T)/d(1/T)]|V is also equal to expression B Homework Equations A: temperature * (dPresure/dTemperature at constant volume) -...

Homework Statement find the largest distance and shortest distance from the origin to the conic whose equation is 6x2 + 4xy +3y2 - 28=0 and hence determine the lengths of the semi axes of this conic Homework Equations Lagrange identity F= f + λφ = 0 distance = d2 =x2+ y2+...

Homework Statement Find the absolute maximum and minimum values of f on the set D. f(x,y) = 1+4x-5y D is the closed triangular region with vertices (0,0) (2,0) (0,3) Homework Equations To find the absolute maximum and minimum values of a continuous function on a closed, bounded set : 1. Find...

Homework Statement I have no homework problem to ask, but rather a general question. Ill give and example of a potential function V = 3x^2 + 2y^2 i know to find Fx i have to differentiate 3x^2 with respect to x and 2y^2 with respect to y. But i have seen cases where someone takes the...

What exactly does it mean for a function to have continuous partial derivatives? How do we see this?

I need help with this one: Find fxy in: ln(x+y)/(xy) .. the ln applies to the whole problem.

I have a question about these two. I have a direction derivative at a in the direction of u defined as: f'(a;u) = lim [t -> 0] (1/t)[f(a + tu) - f(a)] And the partial derivative to be defined as the directional derivative in the direction of u = e_i. My text, Analysis on Manifolds by...

Homework Statement Calculate the following, expressing all results with uncertainties both in absolute and relative (percentage) form: a) A + B b) A x B c) Asin(theta) d) A^2 / Bcos(theta) The relevant formula for the absolute uncertainty is below, but i have no idea how to...

ey guys Generally i just do these without thinking, however i was checking some work today with a friend and he is adament i did my derivative wrong... If i can double check with you Well firstly 'c' is simply a constant q1 and q2 are generalised coordinates IZG1 is simply the...

Homework Statement f(x,y)=2Sin x Cos y g(x,y) = 2Cos x Sin y verify that d(fg)/dx = g(x,y) df/dx + f(x,y) dg/dx The Attempt at a Solution first of all I worked out the partials derivatives in respective to x and y, for both functions df/dx = 2Cos x (but I've a gut feeling that it...

f(x,y)= xy2/(x2+y2) if (x,y)\neq(0,0) =0 if (x,y)=(0,0) Show that the partial derivatives of x and y exist at (0,0). This may be a really stupid question, but would the partial derivatives of x and y at (0,0) just be 0? I tried taking that partial derivatives of xy2/(x2+y2) and...

My professor did this in lecture, and I can't figure out his logic. Can someone fill in the gaps? He went from: dS = \left( \frac{\partial S}{\partial P} \right)_T dP + \left( \frac{\partial S}{\partial T} \right)_P dT (which I totally understand; it just follows from the fact that...

Homework Statement Basically I have two problems that are asking for the partial derivative with respect to x and y at a certain point on a level curve graph, and a contour map. How do you go about doing these? There is no function given, so I don't really know what they expect you to do...

Homework Statement Hi there. Well, I've got some doubts on the partial derivatives for the next function: f(x,y)=sg\{(y-x^2)(y-2x^2)\} Where sg is the sign function. So, what I got is: f(x,y)=f(x)=\begin{Bmatrix}{ 1}&\mbox{ if }& (y-x^2)(y-2x^2)>0\\0 & \mbox{if}& (y-x^2)(y-2x^2)=0\\-1 &...

Homework Statement Hi there. Well, I got the next function, and I'm trying to work with it. I wanted to know if this is right, I think it isn't, so I wanted your opinion on this which is always helpful. f(x,y)=\begin{Bmatrix} (x+y)^2\sin(\displaystyle\frac{\pi}{x+y}) & \mbox{ si }&...

Homework Statement Find the Laplacian of F = sin(k_x x)sin(k_y y)sin(k_z z) Homework Equations \nabla^2 f = \left( \frac{\partial}{\partial x} +\frac{\partial}{\partial y} + \frac{\partial}{\partial z} \right) \cdot \left( \frac{\partial}{\partial x} + \frac{\partial}{\partial y} +...

Hi, Everyone: I was never clear n this point: given that z is a single complex variable, how/why does it make sense to talk about z having partial derivatives.? I mean, if we are given, say, f(x,y); R² -->Rⁿ then it makes sense to talk about...

Hi, As per Clariut's theorem, if the derivatives of a function up to the high order are continuous at (a,b), then we can apply mixed derivatives. I am looking at http://en.wikipedia.org/wiki/Symmetry_of_second_derivatives and I cannot understand in the example for non-symmetry, why the...

Homework Statement Could some mathematically minded person please check my calculation as I am a bit suspicious of it (I'm a physicist myself). This isn't homework so feel free to reveal anything you have in mind. Suppose I have two functions \phi(t) and \chi(t) and the potential V which...

Consider the partial dierential equation, (y4-x2)uxx - 2xyuxy - y2uyy = 1. We will make the substitution x = s2 - t2 and y = s - t, to simplify (a) Solve for s and t as functions of x and y the farthest point i got to was x = s^2 - t^2 = (s+t)(s-t) = y(s+t) y = s - t s+t = x/y i...

Not sure I understand exactly what this question is asking. This is obviously a volume in R3 and so how do you get a tangent inside a volume? Or is it just along the plane y = 2 intersecting the volume? Also, what is a parametric equation...? Thanks for the help: Question: The ellipsoid 4x^2...

My question revolves around the following derivative: for z(x,y) *sorry I can't seem to get the latex right. ∂/∂x (∂z/∂y) What I thought about doing was using the quotient rule to see what I would get (as if these were regular differentials). So, I "factored out" the 1/∂x, then did...

Homework Statement express (\frac{\partial u}{\partial s})_{v} in terms of partial derivatives of u(s,t) and t(s,v) Homework Equations The Attempt at a Solution I'm pretty stuck with this problem. I know that dv = (\frac{\partial v}{\partial s})_{t} ds + (\frac{\partial...

In the midst of https://www.physicsforums.com/showthread.php?t=403002", I came upon a rather interesting, though probably elementary, question. Analagous to the fundamental theorem of calculus, is there a formula or theorem concerning the expression \frac{\partial}{\partial...

When I am taking a partial derivative of an equation with respect to theta_dot, then theta is constant, right? What if I am taking partial derivative with respect to theta, will theta_dot be constant? In this case, theta_dot = omega (angular velocity), but I must keep equation in terms of...

Not a homework question, but It will help me none the less, In my book it says \frac{d}{dt} \int_{-\infty}^{\infty} |\Psi(x,t)|^2 dx is equivalent to \int_{-\infty}^{\infty} \frac{\partial}{\partial t}|\Psi(x,t)|^2 dx I understand how It becomes a partial derivative, since I'm...

Hi, I'm having trouble understanding how people can make calculations using the partial derivatives as basis vectors on a manifold. Are you allowed to specify a scalar field on which they can operate? eg. in GR, can you define f(x,y,z,t) = x + y + z + t, in order to recover the Cartesian...

Homework Statement Suppose that z=f(ax+by), where a and b are constants. Show that bz(x) = az(y). z(x) means partial derivative of z with respect to x, as for z(y). Homework Equations The Attempt at a Solution Say z=ax+by z(x) = a z(y) = b So bz(x) = ba = ab = az(y)...

Do I need to use Schwarz's or Young's theorems to prove it, if don't then do I need to evaluate them on (0,0) using definition.

Homework Statement Prove (∂V/∂T)_s/(∂V/∂T)_p = 1/1-(gamma) (gamma = Cp/Cv) Homework Equations (∂V/∂T)_s = -C_v (kappa)/(beta)T (where beta = 1/V(∂V/∂T)_p, kappa = -1/V(∂V/∂P)_T C_v= - T(∂P/∂T)_v(∂V/∂T)_s The Attempt at a Solution As part(a) ask me to find C_v, I do it similar for...

Homework Statement Suppose the differentiable function f(x,y,z) has the partial derivatives fx(1,0,1) = 4, fy(1,0,1) = 1 and fz(1,0,1) = 0. Find g'(0) if g(t) = f(t2 + 1, t2-t, t+1).Homework Equations The Attempt at a Solution Ok I'm given the solution for this and I'm trying to work through it...

Homework Statement I am translating the question from another language so it might not be word to word with the original question. assume x(s,t) and y(s,t) determined by these two functions: sin(xt) +x+s=1 eyt+y(s+1)=1 The following function is defined H(x,y)=x2+y2-xy such that...

If I have u = u(x,y) and let (r, t) be polar coordinates, then expressing u_x and u_y in terms of u_r and u_t could be done with a system of linear equations in u_x and u_y? I get: u_r = u_x * x_r + u_y * y_r u_t = u_x * x_t + u_y * y_t is this the right direction? Because by...

Hi, so I'm trying to solve Laplace's equation by separation of variables, and there's a basic step I'm not understanding with regards to the product rule. Given A product rule (i think) is taken to make the first term easier to deal with and we get I'm just having trouble...

Homework Statement Given Cartesian coordinates x, y, and polar coordinates r, phi, such that r=\sqrt{x^2+y^2}, \phi = atan(x/y) or x=r sin(\phi), y=r cos(\phi) (yes, phi is defined differently then you're used to) I need to find \frac{d\phi}{dr} in terms of \frac{dy}{dx} Homework...

Homework Statement A mapping f from an open subset S of Rn into Rm is called smooth if it has continuous partial derivatives of all orders. However, when the domain S is not open one cannot usually speak of partial derivatives. Why? Homework EquationsThe Attempt at a Solution In the 1...

Hello, I should feel ashamed to ask this, but it's giving me (and others) some troubles. given f(x_1,\ldots,x_n), is it wrong to say that: \frac{\partial f}{\partial f}=1 ...?

Hi all, I'm looking at the following problem: Suppose that f:\mathbb{R}^2\to\mathbb{R} is such that \frac{\partial{f}}{\partial{x}} is continuous in some open ball around (a,b) and \frac{\partial{f}}{\partial{y}} exists at (a,b): show f is differentiable at (a,b). Now I know that if both...

Homework Statement Find the point on 2x + 3y + z - 11 = 0 for which 4x^2 +y^2 +z^2 is a minimum Homework Equations The Attempt at a Solution Using lagrange multipliers I find: F = 4x^2 + y^2 + z^2 + l(2x + 3y + z) Finding the partial derivatives I get the three equations...

In differential geometry what does df mean as in \mbox{f} : \mathbb{R}^m \mbox{ to } \mathbb{R}^n Then df is what? the jacobian matrix of partial derivatives?

Hello, I'm a student of applied mathematics to economics. Basic course consists of all pure math subjects. We were talking about app's of differentiating the functions u:\mathbb{R}^{n}\to\mathbb{R}^m. We defined a gradient too. In my notes is written: Gravitational potential is a function...

Homework Statement See attatched image. Homework Equations I just don't know where to start... The Attempt at a Solution Any help would be appreciated! :)

Homework Statement 1.if the derivative of f(x,y) with respect to x and y both exist, then f is differentiable at (a,b) 2. if (2,1) is a critical point of f and fxx (2,1)* fyy (2,1) < (fxy (2,1))^2, then f has a saddle point at (1,2) 3. if f(x,y) has two local maxima, then f must have a local...

Hi. So I'm reading a physics book and I come across the following passage: Ok, up to this point I'm fairly confident I'm following along. But then they do the following: and I have no idea where this comes from. I am guessing here that p_i=\phi _i(q) is only in some sufficiently small...

I've been trying to prove that if the following statement holds for all (x,y)ER^2, f must be a linear function: f(x,y)-f(0,0)=x*(d/dx)[f(x,y)]+y*(d/dy)[f(x,y)] It seems to work for any function I plug in, but I'm unable to establish why this always works. Also, when I say (d/dx)[f(x,y)], I...

Homework Statement Let T= g(x,y) be the temperature at the point (x,y) on the ellipse x=2sqrt2 cos(t) and y= sqrt2 sin(t), t is from 0 to 2pi. suppose that partial derivative of T with respect to x is equal to y and partial derivative of T with respect to y is equal to x. Locate the max and...