Partial Definition and 1000 Threads

For a conservative force \vec{F}=-\vec{\nabla} U \implies dW=-\vec{\nabla}U \cdot d\vec{s} Where d\vec{s} is the infinitesimal vector displacement. Does the following hold? -\frac{\partial U}{\partial \vec{s}}=-\vec{\nabla} U \cdot d\vec{s}=d W, i.e. the infinitesimal work is minus the...

Homework Statement Suppose ω = g(u,v) is a differentiable function of u = x/y and v = z/y. Using the chain rule evaluate $$x \frac{\partial ω}{\partial x} + y \frac {\partial ω}{\partial y} + z \frac {\partial ω}{\partial z}$$ Homework EquationsThe Attempt at a Solution u = f(x,y) v = h(y,z)...

Homework Statement \frac{d}{dt}\left(\frac{\dot{q}}{\sqrt{1+\left(\dot{q}\right)^{2}}}\right)=0\Rightarrow\frac{\dot{q}}{\sqrt{1+\left(\dot{q}\right)^{2}}}=const\Rightarrow\dot{q}=A\Rightarrow q=At+B Homework Equations Why it ok to say that...

Homework Statement v_{i}=\dot{x}_{i}=\dot{x}_{i}\left(q_{1},q_{2},..,q_{n},t\right) T \equiv \frac{1}{2}\cdot{\sum}m_{i}v_{i}^{2} \frac{\partial T}{\partial\dot{q}_{k}}={\sum}m_{i}v_{i}\frac{\partial v_{i}}{\partial\dot{q}_{k}}={\sum}m_{i}v_{i}\frac{\partial x_{i}}{\partial q_{k}}[/B]...

Homework Statement ∂z/∂x of ycos(xz)+(4xy)-2z^2x^3=5x[/B] Homework Equations n/a The Attempt at a Solution ∂z/∂x=(5+yz-4y+6z^2x^2)/(-yxsin(xz)-4zx^3)[/B] Is this correct? Just trying to make sure that's the correct answer. I appreciate the help. I can post my work if need be. Thanks

Homework Statement The question is stated at the top of the attached picture with a solution 20160303_095831.jpg The correct results of the coefficients are A=2, B=-5, C=1 I have tried this problem multiple times and am still getting ugly coefficients. I have no idea why. A fresh pair of eyes...

I'm sure you are all familiar with calculating the inductance of a long transmission line. We first calculate the partial self inductance and we add to the partial mutual inductance due to the current in the other conductors. Looking at the image of a single-phase system, where I1 + I2 = 0...

I'm trying to find the partial derivatives of: f(x,y) = ∫ (from -4 to x^3y^2) of cos(cos(t))dt and I am completely lost, any help would be appreciated, thanks.

Sorry about the title, had a hard time trying to fit the question on the given space. The question is quite simple : If F = F(x_1,...,x_n,t) , Under what conditions is \frac{d }{dt} \frac{\partial F }{\partial xi} = \frac{\partial }{\partial xi} \frac{dF }{dt} true?

(Hope it's okay that I'm posting so much at the moment, I'm having quite a bit of trouble with something I'm doing) Homework Statement I'm having trouble with the simplification of the following equation. The answer is shown, but I can't figure out the process to get to it. \frac{d}{dt}...

Homework Statement [/B] The area of a triangle is (1/2)absin(c) where a and b are the lengths of the two sides of the triangle and c is the angle between. In surveying some land, a, b, and c are measured to be 150ft, 200ft, and 60 degrees. By how much could your area calculation be in error if...

What does it mean when it says to classify the second-order partial differential? (See attached) How would I get started?

Homework Statement the equation is E= k((xy)x[hat] +(2yz)y[hat] +(3xz)z[hat]) Homework Equations partial of x with respect to y on the x component partial of y with respect to x on the y component The Attempt at a Solution my professor said during class that the partial of x with respect to y...

Hey guys, so my professor told me to take a look at an equation, because he thinks that there is a mistake. We are basically talking about exercise 6.3 (on last image). The pictures will show you the text, so that you have all the information, that I have http://puu.sh/mrNDl/ec19cdff63.png...

In my book, applied analysis by john hunter it gives me a strange way of stating an infinite sum that I'm still trying to understand because in my calculus books it was never described this way. It says: We can use the definition of the convergence of a sequence to define the sum of an...

Hello everybody! I have to decompose to simple fractions the following function: $$V(z)=\frac{z^2-4z+4}{(z-3)(z-1)^2}$$. I know I can see the function as: $$V(z)=\frac{A}{z-3}+\frac{B}{(z-1)^2}+\frac{C}{z-1}$$, and that the terms A, B, C can be calculated respectively as the residues in 3...

Homework Statement Decompose \frac{2(1-2x^2)}{x(1-x^2)} I get A = 2, B =-1, C = 1, but this doesn't recompose into the correct equation, and the calculators for partial fraction decomposition online all agree that it should be A = 2, B = 1, C = 1. Here is one of the online calculator results...

take the function f(x,y,z) s.t dF=(d'f/d'x)dx+(d'f/d'y)dy+(d'f/d'z)dz=0 where "d'" denotes a curly derivative arrow to show partial derivatives Mod note: Rewrote the equation above using LaTeX. $$df = (\frac{\partial f}{\partial x} ) dx + (\frac{\partial f}{\partial y} ) dy + (\frac{\partial...

Mark44 submitted a new PF Insights post Partial Fractions Decomposition Continue reading the Original PF Insights Post.

I've just done a derivation and had to use the following u_{b}u^{c}\partial_{c}\rho = u_{b}\frac{dx^{c}}{d\tau}\frac{\partial\rho}{\partial x^{c}} = u_{b}\frac{d\rho}{d\tau} We've done this cancellation a lot during my GR course, but I'm not clear exactly when/why this is possible. EDIT: is...

Hi all, I know that the dimension of a partial decay width or a cross section should be GeV or pb respectively. But what if i have a decay width probational to ## \Gamma = 10^{-3} GeV^3 G_\mu ## where I calculated all the masses and constants in ## \Gamma ##, ## G_\mu ## is the Fermi...

Homework Statement Given n=(x + iy)/2½L and n*=(x - iy)/2½L Show that ∂/∂n = L(∂/∂x - i ∂/∂y)/2½ and ∂/∂n = L(∂/∂x + i ∂/∂y)/2½ Homework Equations ∂n Ξ ∂/∂n, ∂x Ξ ∂/∂x, as well as y. The Attempt at a Solution ∂n=(∂x + i ∂y)/2½L Apply complex conjugate on right side, ∂n=[(∂x + i ∂y)/2½L] *...

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...

Homework Statement integrate (4x+3)/(x^2+4x+5)^2 Homework EquationsThe Attempt at a Solution I know to solve this problem you have to work with partial fractions, in the solution we were given they solve as followed 4x+3=A(x^2+4x+5)'+B I don't know why they take the derivative of x^2+4x+5...

x2y2 + (y+1)e-x=2 + x Defines y as a differentiable function of x at point (x, y) = (0,1) Find y′: My attempt: ∂y/∂x =2xy3 + (-y-1)e-x=1 ∂y/∂y = 3x2y2 - e-x=0 Plugging in for x and y ⇒ ∂y/∂x = -3 ∂y/∂x = -1 For some reason I think y′ is defined as (∂y/∂x) /(∂y/∂y) = 3 At leas this give...

I have a question where f(x) = 20-2x^2/(x-1)(x+2)^2 and have solved for constants A,B and C. A = 2 B = -4 C = -4 I have worked this out myself. Now I am told to compute the indefinite integral and I am getting this answer but apparently it is wrong and I don't understand how? My answer...

We have a function: ##\phi'(t',x')##. We want to find: ##\frac{\partial\phi'}{\partial x}##. I know that the answer is: ##\frac{\partial\phi'}{\partial x} = (\frac{\partial\phi'}{\partial t'} \cdot \frac{\partial t'}{\partial x}) + (\frac{\partial\phi'}{\partial x'} \cdot \frac{\partial...

Is my background enough to learn partial differential equations? I have completed up to calculus 2 and linear algebra. I am currently taking Cal 3 and Ordinary Differential Equations. I am doing well in both courses. I would like to learn PDE and a bit more Linear Algebra, during the winter...

F(r,s,t,v) = r^2 + sv + t^3, where: r = x^2 +y^2+z^2 /// s = xyz /// v = xe^y /// t = yz^2 find Fxx i have 2 solutions for this and i am not sure what is the right one the first solution finds Fx then uses formula : Fxx = Fxr.Rx + Fxs.Sx + Fxv.Vx+Fxt.Tx the 2nd solution find Fx then uses the...

Hello Is what I have done calculated here correct? Please correct me if I have done something wrong. Thanks in advance.

so last night I get on a sleep number bed and the remote reads 35 (unitless - I am assuming this number is related to pressure.) I click it down once to 30 and it deflates nearly completely. I get off the mattress, the reading drops to 5 or 10 and the mattress begins to inflate to 30. So this...

Homework Statement Homework Equations Chain rule, partial derivation The Attempt at a Solution dv/dt=dv/dx*dx/dt+dv/dy*dy/dt dx/dt=-4t -> evaluate at (1,1) =-4 dv/dt=-4dv/dx+4(-2) dv/dt=-4dv/dx-8 How can I find the missing dv/dx in order to get a value for dv/dt? Thanks!

Hello I'm currently trying to solve these two problems: 1) Find the partial derivatives ∂m/∂q and ∂m/∂h of the function: m=ln(qh-2h^2)+2e^(q-h^2+3)^4-7 Here, I know I should differentiate m with respect to q while treating h as a constant and vice versa. But I'm still stuck, and I'm not sure...

\begin{array}{l} u = u(x,y) \\ v = v(x,y) \\ and\\ {u_x} + 4{v_y} = 0 \\ {v_x} + 9{u_y} = 0 \\ with\ the\ initial\ conditions \\ u(x,0) = 2x _(3)\\ v(x,0) = 3x _(4)\\ \end{array} Easy, u_{xx}-36u_{yy}=0 and v_{xx}-36v_{yy}=0 General solution u\left ( x,y \right )=h\left ( x+6y \right...

I was looking over a derivation to find the laplacian from cartesian to cylindrical and spherical coordinates here: http://skisickness.com/2009/11/20/ Everything seems fine, but there is an instance (I have attached a screenshot) where implicit differentiation is done to find $$ \frac {\partial...

Let ##f(x,y)## be a scalar function. Then $$\frac{∂f}{∂x} = \lim_{h \rightarrow 0} \frac{f(x+h,y)-f(x,y)}{h} = f_x (x,y)$$ and $$\frac{∂}{∂y} \left (\frac{∂f}{∂x} \right ) = \lim_{k \rightarrow 0} \frac{f_x(x,y+k)-f_x(x,y)}{k} = \lim_{k \rightarrow 0} \left ( \frac{ \displaystyle \lim_{h...

I am working on implementing a PDE model that simulates a certain physical phenomenon on the surface of a 3D mesh. The model involves calculating mixed partial derivatives of a scalar function defined on the vertices of the mesh. What I tried so far (which is not giving good results), is this...

Let the PDE $u_{xx}-4u_{xy}+4u_{yy}=0.$ Reduce to the canonical form.Good Morning MHB :). My problem is find the canonical form of the PDE know an variable change. But how I can transform the equation? Thanks.

Find surface of $\begin{array}{l} \text{Problem Cauchy} \\ {a^2} \cdot {x_2} \cdot u \cdot {u_{{x_1}}} + {b^2} \cdot {x_1} \cdot u \cdot {u_{{x_2}}} = 2{c^2}{x_1}{x_2}{\rm{ }} \\ \end{array}$ The partial differntial equation passes through ${\rm{ C: = \{ }}\frac{{{x^2}}}{{{a^2}}} +...

What is the difference between partial derivatives and gradients, if there is any? I'm asking because I need to derive a function " f (T,P) " for air convection; where T is temperature and P is pressure and both are variables in this case. Thanks

Homework Statement Let l, w, and h be the length, width and height of a rectangular box. The length l is increasing with time at at rate of 1 m/s, while the width and the height are decreasing at rates 2 m/s and 1m/s respectively. At a certain moment in time the dimensions of the box are l=5...

I have a multivariable function z = x2 + 2y2 such that x = rcos(t) and y = rsin(t). I was asked to find (I know the d's should technically be curly, but I am not the best at LaTeX). I thought this would just be a simple application of chain rule: ∂2/(∂y∂t) = (∂z/∂x)(ⅆx/ⅆt) + (∂z/∂y)(ⅆy/ⅆt)...

Homework Statement Find the Inverse laplace transform of: http://www4c.wolframalpha.com/Calculate/MSP/MSP14541hg721e74730d4fb00004644i96f59549h1d?MSPStoreType=image/gif&s=30&w=201.&h=40. Result...

From the resonance structures of Naphthalene, 1-2 Bond has more double Bond character than 2-3 Bond. In March's Advanced organic chemistry, its given that ozone preferentially reacts with 1-2 Bond. But the reaction is not given. Is this a normal ozonolysis reaction in which the 1-2 Bond is...

Homework Statement What are the partial half of 22Na for decay by a)Ec b) β+ emission Homework Equations λ=ln2/T1/2 The Attempt at a Solution this what I do T1/2 =2.602 Yr λ=ln2/2.602 λ=0.266 yr-1what is the difference between a)Ec b) β+ emission there is no Percentage of each decay type.!

Hi, I'm a little confused about something. I have an object, and I want to take the partial derivative of its position wrt velocity and vice versa. I'm not sure how to begin solving this problem. Essentially, what I have is this: ## \frac{\partial x}{\partial \dot x} ## and ## \frac{\partial...

Homework Statement Find the equation of state given that k = aT^(3) / P^2 (compressibility) and B = bT^(2) / P (expansivity) and the ratio, a/b? Homework Equations B = 1/v (DV /DT)Pressure constant ; k = -1/v (DV /DP)Temperature constant D= Partial derivative dV = BVdT -kVdP (1) ANSWER is...

Homework Statement Homework Equations trigonometric identities The Attempt at a Solution I did a trig substitution of u=tan(θ/2) and from that I could substitute cos(θ) = 1-u2/1+u2 dθ = 2/(1+u2) du = 1/2 sec2(θ/2) dθ I simplified a bit and changed the bounds to get 2du/(5u2 + 1)(1 + u2)2...

Hey all, I am reading Goldstein and I am at a point where I can't follow along. He has started with D'Alembert's Principle and he is showing that Lagrange's equation can be derived from it. He states the chain rule for partial differentiation: \frac{d\textbf{r}_i}{dt}=\sum_k \frac{\partial...

Mod note: Moved from a homework section 1. Homework Statement N/A Homework Equations f(x + Δx,y) = f(x,y) + ∂f(x,y)/∂x*Δx The Attempt at a Solution Sorry this isn't really homework. We were given this equation today in order to derive the Taylor expansion formula in two variables and I'm not...