Partial derivatives Definition and 435 Threads

In the solution to a differential-equation problem -- proof of the existence of an integrating factor -- the following statements are made regarding a general function "u(xy)" [that is, a function of two variable that depends exclusively on the single factor "x*y"]...

Homework Statement Last week we've been doing partial derivatives and I understood all that , but also skipped last lecture.I asked colleague about the lecture and he told me that professor mentioned something about rotor and he gave an example of such and such problem.Anyway here it goes...

Homework Statement Prelim: my question is about a very specific part of a question whereby the student is asked to derive the final formula for the general solution in two vars, but I will post the entire question for clarify. Newton's Method for approximating the roots of an equation f(x)=0...

Homework Statement Let f\colon\mathbb{R}^m\to\mathbb{R}. All partial derivatives of f are defined at point P_0\colon = (x_1, x_2, ... , x_m). If f has local extremum at P_0 prove that \frac{\partial f}{\partial x_j} (P_0) = 0, j\in \{1, 2, ..., m\} Homework Equations Fermat's theorem: Let...

Homework Statement a) Show that the function f(x,y)=\sqrt[3]{xy} is continuous and the partial derivatives f_x and f_y exist at the origin but the directional derivatives in all other directions do not exist b) Graph f near the origin and comment on how the graph confirms part (a). 2. The...

When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##? \frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...

Homework Statement Show that the coefficient of volume expansion can be expressed as β= -1÷ρ (∂ρ÷∂T) keeping P (pressure) constant Where rho is the density T is Temperature Homework Equations 1/v =ρ β= 1/v (∂v÷∂T) keeping P (pressure ) constant The Attempt at a Solution I started with...

Homework Statement Show that if f is homogeneous of degree n, then x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf(x,y) Hint: use the Chain Rule to diff. f(tx,ty) wrt t. 2. The attempt at a solution I know that if f is homogeneous of degree n then t^nf(x,y) =...

If three variables x,y and z are related via some condition that can be expressed as $$F(x,y,z)=constant$$ then the partial derivatives of the functions are reciprocal, e.g. $$\frac{\partial x}{\partial y}=\frac{1}{\frac{\partial y}{\partial x}}$$ Is the correct way to prove this the following...

Homework Statement Write the chain rule for the following composition using a tree diagram: z =g(x,y) where x=x(r,theta) and y=y(r,theta). Write formulas for the partial derivatives dz/dr and dz/dtheta. Use them to answer: Find first partial derivatives of the function z=e^x+yx^2, in polar...

Hi, I'm trying to wrap my head around the derivation of the wave equation and wave speed. For starters I'm looking at the derviation done on this site: http://www.animations.physics.unsw.edu.au/jw/wave_equation_speed.htm I could maybe explain what I understand at this point Given a string with...

Homework Statement let w(u,v) = f(u) + g(v) u(x,t) = x - at v(x,t) = x + at show that: \frac{\partial ^{2}w}{\partial t^{2}} = a^{2}\frac{\partial ^{2}w}{\partial x^{2}} The Attempt at a Solution w(x-at, x+at) = f(x-at) + g(x+at) \frac{\partial }{\partial t}(\frac{\partial w}{\partial...

Homework Statement Let ##C## be a level curve of ##f## parametrized by t, so that C is given by ## x=u(t) ## and ##y = v(t)## Let ##w(t) = g(f(u(t), v(t))) ## Find the value of ##\frac{dw}{dt}## Homework Equations Level curves Level sets Topographic maps The Attempt at a Solution Is it true...

Homework Statement show that the following functions are differentiable everywhere and then also find f'(z) and f''(z). (a) f(z) = iz + 2 so f(z) = ix -y +2 then u(x,y) = 2-y, v(x,y) = x Homework Equations z=x+iy z=u(x,y) +iv(x,y) Cauchy-Riemann conditions says is differentiable everywhere...

Homework Statement x^2 + y^2 < 1 Find the partial derivatives of the function. Homework Equations x^2 + y^2 < 1 The Attempt at a Solution @f/@x = 2x = 0 @f/@y = 2y = 0 4. Their solution @f/@x = 2x = 0 @f/@y = 2y + 1 = 0 5. My Problem I don't see how / why they get 2y + 1 for the...

Homework Statement Homework Equations included in the first picture The Attempt at a Solution i feel confident in my answer to part "a". i pretty much just did what the u and v example at the top of the page did. but for part "b" i tried to distribute and collect like terms and what not...

Hey, Little confused by something: if we have u=x+y and v=xy what is the partial derivative w.r.t. u of y^2=uy-v I am told it is 2y (dy/du) = u (dy/du) + y And I can see where these terms come from. What I don't understand is why there is no (dv/du) term, as v and u aren't independent...

Recently I started with multivariable calculus; where I have seen concepts like multivariable function, partial derivative, and so on. A week ago we saw the following concept: directional derivative. Ok, I know the math behind this as well as the way to compute the directional derivative through...

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...

Homework Statement I was given the following function f(x,y) = \begin{cases} \frac{x^2y}{x^4+y^2} & (x,y) \neq 0 \\ 0 & (x,y) = 0 \end{cases} Which of the following are true? (I) f is not continuous at (0, 0). (II) f is differentiable everywhere (III) f as a well defined partial...

Homework Statement Find \frac{\partial f}{\partial x} if f(x,y)=\cos(\frac{x}{y}) and y=sinx Homework Equations See above The Attempt at a Solution For \frac{\partial f}{\partial x} I calculated -\frac{1}{y}\sin(\frac{x}{y}) which comes out as \frac{-\sin(\frac{x}{\sin(x)})}{sinx} and this...

x^2 - y^2 +2mn +15 =0 x + 2xy - m^2 + n^2 -10 =0 The Question is: Show that del m/ del x = [m(1+2y) -2 x n ] / 2 (m^2 +n^2) del m / del y = [x m+ n y] / (m^2 +n^2) note that del= partial derivativesMy effort on solving this question is Fx1=2x Fm1=2n Fx2 =2y Fm2 =-2m del m /del x =...

Hi all, When you have a surface defined by F(x, y, z) = 0 where x = f(t), y= g(t) and z= h(t) and a point on this surface P_0 = (x_0, y_0, z_0) , could someone explain to me why a line through P_0 with direction numbers [\frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt}] is perpendicular to a...

Hello PF! It's been a while since I last posted here. I have come across a problem in my textbook, which asks me to find expressions for V as a function of T and P, starting from the coefficients of thermal expansion and compressibility. \alpha = \frac{1}{V} \left(\frac{\partial V}{\partial T}...

I know how to prove this via limits and I'm okay with that. What I want to understand is the interpretation of the theorem and specifically a visualisation of why what the theorem states must be the case. My guess is that this theorem is saying that change is symmetrical. But I don't know...

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...

Hello. Can someone check if my partial derivatives are correct? I am not so confident with my answers.

Given V=xf(u) and u = \frac{y}{x} How do you show that: x^2 \frac{\partial^2V}{\partial x^2} + 2xy\frac{\partial^2V}{\partial x\partial y} + y^2 \frac{\partial^2V}{\partial y^2}= 0 My main problem is that I am not sure how to express V in terms of a total differential, because it is a...

Homework Statement Suppose z=ψ(2x-3y), Show that the second partial derivative of z with respect to x, is equal to the second partial derivative with respect to y multiplied by a scalar k. Homework Equations The Attempt at a Solution I thought this was too simple to be correct...

Suppose we have a function V(x,y)=x^2 + axy + y^2 how do we write \frac{dV}{dt} For instance if V(x,y)=x^2 + y^2, then \frac{dV}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt} So, is the solution \frac{dV}{dt} = 2x \frac{dx}{dt} + ay\frac{dx}{dt} + ax\frac{dy}{dt} + 2y \frac{dy}{dt}

Hi guys, Question is: Find the slopes of the curves of intersection of surface z = f(x,y) with the planes perpendicular to the x-axis and y-axis respectively at the given point. z = 2x2y ...at (1,1). fx(x,y) = 4xy ∴ Slope = 4 fy(x,y) = 2x2 ∴ Slope = 2 Is this wrong? Answer...

Hello all, I have this function here: \[f(x,y)=\left\{\begin{matrix} z &(x,y)\neq (0,0) \\ 0 & (x,y)=(0,0) \end{matrix}\right.\] where \[z=\frac{x^{3}+xy^{2}}{2x^{2}+y^{2}}\] And I need to find it's first partial derivative by x and y at the point (0,0). I am not sure I know how to approach...

Homework Statement Using the formal limit definition of the derivative, derive expressions for the Fourier Transforms with respect to x of the partial derivatives \frac{\partial u}{\partial t} and \frac {\partial u}{\partial x} . Homework Equations The Fourier Transform of a function...

Good afternoon guys! I have some doubts about partial derivatives. The other day, my analytic geometry professor told us that slopes do not exist in three-dimensional space. If that's the case, then what does a partial derivative represent? Given that the derivative of a function with respect to...

Hello! :) Having the transformations: $$\xi=\xi(x,y), \eta=\eta(x,y)$$ I want to find the following partial derivatives: $$\frac{\partial}{\partial{x}}= \frac{\partial}{ \partial{\xi}} \frac{\partial{\xi}}{\partial{x}}+\frac{\partial}{\partial{\eta}}...

Problem: I did some of the problem on MatLab but I'm having a difficult time evaluating the derivatives at (0,0). Also, MatLab gave me the same answer for fxy and fyx, which according to the problem isn't correct. Any ideas? I used MatLab and computed: fx(x,y)=(2*x^2*y)/(x^2 + y^2) + (y*(x^2 -...

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.

Homework Statement If ##xs^2 + yt^2 = 1## (1) and ##x^2s + y^2t = xy - 4,## (2) find ##\frac{\partial x}{\partial s}, \frac{\partial x}{\partial t}, \frac{\partial y}{\partial s}, \frac{\partial y}{\partial t}## at ##(x,y,s,t) = (1,-3,2,-1)##. Homework Equations Pretty much those just listed...

Hello all, I am trying to calculate the second order of the partial derivative by x of the function: f(x,y)=(x^2)*tan(xy) In the attach images you can see my work. Both the answer in the book where it came from and maple say that the answer is almost correct, but not entirely. In the last...

Ok folks, I've taken a stab at the Latex thing (for the first time, so please bear with me). I've mentioned before that I'm teaching myself relativity and tensors, and I've come across a question. I have a few different books that I'm referencing, and I've seen them present the ordinary...

Hello, I am not completely certain why in thermodynamics, it seems that everything is done as a partial derivative, and I am wondering why? My guess is because it seems like variables are always being held constant when taking derivatives of certain things, but it is still somewhat a mystery to...

Hi I have a question about partial derivatives? For example if I have a function x = r cos theta for all functions, not just for this function will dx/d theta be the inverse of dtheta/dx, so 1 divided by dx/d theta will be d theta/ dx? Please help on this partial derivative question...

Homework Statement let u=f(x,y) , x=x(s,t), y=y(s,t) and u,x,y##\in C^2## find: ##\frac{\partial^2u}{\partial s^2}, \frac{\partial^2u}{\partial t^2}, \frac{\partial^2u}{\partial t \partial s}## as a function of the partial derivatives of f. i'm not sure I'm using the chain rules...

Se a function f(x(t, s), y(t, s)) have as derivative with respect to t: \frac{df}{dt}=\frac{df}{dx} \frac{dx}{dt}+\frac{df}{dy} \frac{dy}{dt} And, with respect to s: \frac{df}{ds}=\frac{df}{dx} \frac{dx}{ds}+\frac{df}{dy} \frac{dy}{ds} But, how will be the derivative with respect to...

The question is: a) Find explicit expressions for an ideal gas for the partial derivatives: (∂P/T)T, (∂V/∂T)P and (∂T/∂P)V b) use the results from a) to evaluate the product (∂P/V)T*(∂V/∂T)P*(∂T/∂P)V c) Express the definitions of V(T,P) KT(T,P)an BT(T,V) in terms of the indicated independent...

I am given Z = f (x, y), where x= r cosθ and y=r sinθ I found ∂z/∂r = ∂z/∂x ∂x/∂r + ∂z/∂y ∂y/∂r = (cos θ) ∂z/∂x + (sin θ) ∂z/∂y and ∂z/∂θ = ∂z/∂x ∂x/∂θ + ∂z/∂y ∂y/∂θ= (-r sin θ) ∂z/∂x + (r cos θ) ∂z/∂y I need to show that ∂z/∂x = cos θ ∂z/∂r - 1/r * sin θ ∂z/∂θ and ∂z/∂y = sin...

Hi, I am sort of hung up with a particular step in a derivation, and this has caused me to ponder a few properties of partial derivatives. As a result, I believe I may be correct for the wrong reasons. For this example, the starting term is (\frac{\partial}{\partial x}\frac{\partial...

Is there a standard notation for partial derivatives that uses indexes instead of letters to denote ideas such as the 3 rd partial derivative with respect to the the 2nd argument of a function? As soon as a symbol gets superscripts and subscripts like \partial_{2,1}^{3,1} \ f the spectre of...

Partial Derivatives Hi all I was wondering if anyone could help me with this problem. I have a triangle that has a = 13.5m, b = 24.6m c, and theta = 105.6 degrees. Can someone remind me of what the cosine rule is? Also (my question is here) From the cosine rule i need to find: the...

Hi Everyone, I was studying coordinate transformation and I came across this equation, that I couldn't understand how it came up. Let me put it this way: x = rcosθ Then if I want to express the partial derivative (of any thing) with respect to x, what would be the expression? i.e. ∂/∂x=...