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Questions tagged [partial-derivative]

For questions regarding partial derivatives. The partial derivative of a function of several variables is the derivative of the function with respect to one of those variables, with all others held constant.

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29 views

Finding absolute minimum and maximum using lagrange mutlipliers?

I have this function $f(x,y)=3y^2-12x^2+1$ with the constraint $h(x,y)=x^2+y^2-x-2y+\frac54=0$. First thing I do is write $$3y^2-12x^2+1-λ(x^2+y^2-x-2y+\frac54)$$ which expands to $$3y^2-12x^2+1-...
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1answer
14 views

Let $f (z) = u+iv$ be an analytic function, then show that $ (∂^2/∂x^2 + ∂^2/∂y^2)|f(z)|^2 = 4|f'(z)|^2 $.

Let $f (z) = u+iv$ be an analytic function, then show that $$(∂^2/∂x^2 + ∂^2/∂y^2)|f(z)|^2 = 4|f'(z)|^2\,.$$ $f(z) = u + iv $ $ϕ = |f(z)|^2 = u^2 + v^2 $ $f'(z) = ∂u/∂x + i∂v/∂x $ $|f'(z)|^2 = (∂u/...
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1answer
20 views

Minimizing the sum of distances between points and a point on the plane?

This seems like it should be a simple problem, so maybe I am just being silly. Let's say that we have $n$ points, $P_1$ to $P_n$, with coordinates $(x_1,y_1, z_1)$ to $(x_n,y_n, z_n)$, floating above ...
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0answers
15 views

Twice dericative formula in spherical coordinates

Is there a way to easily prove the following formula? $\nabla^2 f=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2\frac{\partial f}{\partial r})+\frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}(...
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1answer
18 views

extrema of $z = y/x$ and $z = ye^{x^2}$

I am trying to find the extrema of these two functions and classify them. a: $z = \frac{y}{x}$ b: $z = ye^{x^2}$ However, when I evaluate their gradients, I find that they are never equal to 0 or ...
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2answers
52 views

Why is dx/dt = -(∂u/∂t) / (∂u/∂x)? [on hold]

I found that $\frac{dx}{dt} = -\cfrac{ \frac{\partial u}{\partial t} }{ \frac{\partial u}{\partial x}}$ on the internet. I can´t figure out if it is true and why.
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20 views

Rules for inverse functions and partial derivatives

I have that $u(t,x)$ satisfies $\partial u/\partial t + u \cdot \partial u/\partial x = 0$ I need to show that if $x = x(t)$, then $dx/dt = u(t,x)$ So far I have $u = -\partial u/\partial t \...
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1answer
17 views

Gradient direction

I learned at the math classes that the gradient at a point is perpendicular to the surface but our electromagnetism teacher taught us that the gradient is tangent to the graph and it points in the ...
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40 views

Does existence of mixed derivative imply continuity?

I would like to does existence of mixed derivative at a point guarantees the function is continuous at that point? And I would like to ask if ${\frac {\partial ^{2}f}{\partial x^{2}}}$ exist implies ...
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$(-v+u)u_ξ = u_{ξξ}$ $\rightarrow$ $\frac{\partial}{\partial_ξ}(u_ξ - vu + \frac{1}{2} u^2) =0$

$(-v+u)u_ξ = u_{ξξ}$ $\rightarrow$ $\frac{\partial}{\partial_ξ}(u_ξ - vu + \frac{1}{2} u^2) =0 $ where u=u(ξ,$\tau$) , $t=\tau$ , $x-vt = \epsilon ξ$ and $\frac{\partial u}{\partial t} + u\frac{\...
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3answers
37 views

How to find critical points of two equations?

So i have this function $$f(x, y) = (x − y)(1 − xy)$$ which i re-write as $x-x^2y-y+xy^2$ I find the partial derivatives: $$f_x(x,y)=1-2xy+y^2$$ $$f_y(x,y)=-x^2-1+2xy$$ I want to look for ...
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1answer
18 views

Value of multivariable function

We have $$f(x,y,z)=xz+x^2z+sin(x+2y+z).$$ What is the the value of $df(1,-1,1)$. I found the partial derivatives of f and than what? Is something like $$df(a,b,c,)=\frac{df}{dx}(a,b,c)+\frac{df}{dy}(a,...
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2answers
43 views

Find $\frac{\partial y}{\partial z}$ of the surface $g(s,t)=(s^2+2t,s+t,e^{st})$ near $g(1, 1) = (3, 2, e)$.

Consider the surface given by $g(s, t) = (s^2 + 2t, s + t, e^{st})$. Think of $y$ as a function of $x$ and $z$. Find $\dfrac{\partial y}{\partial z}(3,e)$ near $g(1, 1) = (3, 2, e)$. really ...
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1answer
49 views

Chain rule when applying L'Hopital's rule

I have a very basic question regarding derivation function: $$f(\omega(t)) = \frac{2 +x(t)\cdot \frac{d\omega(t)}{dt}}{\omega(t)} $$ when I check for $$= \lim_{\omega(t)\to\ 0}\frac{2 +x(t)\cdot\...
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0answers
52 views

How to prove the derivative of $a^{T} A^{-1} b$ with respect to $A$

So I can find from the matrix cookbook here: $\frac{\partial a^TX^{-1}b}{\partial X} = -X^{-T}ab^TX^{-T}$ To prove it, I have tried expanding: $a^TX^{-1}b = \sum\limits_{i,j}^{n,n}a_i(X^{-1})_{ij}...
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1answer
24 views

Partial derivative of a two variables function, one of which dependent on the other

I found this exercise on the book of multivariable calculus from which I'm studying: "Find the partial derivative $\frac{\partial{z}}{\partial{x}}$ and the total derivative $\frac{\text{d}z}{\text{d}...
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1answer
31 views

Plane tangent to sin(xyz) = x + 2y + 3z

I'm trying to solve the following : Find the tangeant plane to sin(xyz) = x + 2y + 3z at P(2,-1,0) Fx = 1 - cos(xyz)*yz = 1 Fy = 2 - cos(xyz)*xz = 2 Fz = 3 - cos(xyz)*xy = 5 So my gradient would ...
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1answer
18 views

How to find these derivatives of a implicit function.

The equation $x^3 + 2y^3 + z^3 + xyz = 4$ $x,y > 0$ describe a graph of a function: $ z = f(x,y)$. Find, $\frac{\partial f}{\partial x}$ and $ \frac{\partial^2 f}{\partial x\partial y}$ So I ...
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19 views

derivative of trace of inverse matrix which is a function of 1 parameters

I have a trace of inverse matrix which is a function of two parameters and need to find its derivative. $\frac{\partial \sum_{i=1}^{n}B^{T}A_{i}^{-1}\left(t\right) B}{\partial t}=\sum_{i=1}^{n}\frac{\...
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0answers
8 views

PDE with Robin Boundary Condition at alpha Solving with Poisson's Equation

Using a code like this, I am having a hard time applying a Robin Boundary condition for a instead of a dirichlet for the following problem: IMG OF QUESTION FOR POISSON ROBIN BC Nx = ...
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0answers
48 views

Finding $\frac{\partial^2 u}{\partial x \partial y}$ of $u=f(x^2 z, z^2+x, y^2z)$.

$$ A=x^2z,\;B=z^2+x,\;C=y^2z\\ u'_x=f'_A A'_x+f'_B B'_x+f'_C C'_x=f'_A\cdot 2xz+f'_B\\ u''_{xy}=(f'_A\cdot 2xz+f'_B)'_y=f''_{Ay}\cdot 2xz+f''_{By}=2xz\frac{\partial^2 f}{\partial(x^2z)\partial y}+\...
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1answer
11 views

Formula for tangent plane to surface given by parametrization

I am aware of how to find an equation of the tangent place to a surface that is given as the graph of a function $z = g(x,y)$. Here one finds a normal vector by essentially taking the partial ...
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2answers
20 views

The partial derivative

Suppose the variables $x$ and $u$ are related by $$x=u$$ Then I have a function $f=f(x)$ which does not explicitly depend on $u$. Then is it true that $$\frac{\partial f}{\partial u}=0$$?
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reciprocal of the Schwarz theorem

May you help me with a counterexample of the reciprocal of the Schwarz theorem? i.e. a $\mathbb R^2$-function $f$ verifying $\frac{\partial^2 f}{\partial x \partial y} (a) = \frac{\partial^2 f}{\...
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1answer
18 views

Higher Order Multivariable Taylor Expansions

The quadratic multivariable Taylor approximation of a function $f(x, y)$ around a point $(a, b)$ is given by $f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) + \frac{1}{2}f_{xx}(a, b)(x - a)^2 + f_{xy}(...
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1answer
66 views

Derivative of $x^{\beta}$

Let $x\in\mathbb{R}^d$, and $\alpha,\beta\in\mathbb{N}^d$ such that $|\alpha|\in\left\{1,2\right\}$ $|\beta|\ge 3$ What is the result of $$\partial_x^{\alpha}x^{\beta}$$? Thanks
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1answer
20 views

Monotonically increasing functions

In Baby Rudin, Theorem 5.11 says, Suppose $f$ is differentiable in $(a,b)$. If $f'(x) \geq 0$ for all $x \in (a,b)$, then $f$ is monotonically increasing, but this is an if and only if, right? If we ...
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0answers
8 views

Continuity of partial derivative

I come up with this question while working on an exercise. Suppose that $f(x,y)$ is defined in some neighborhood of $(x_0,y_0)$, $\frac{\partial^2 f}{\partial x \partial y}$ exists and is continuous ...
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10 views

$\textbf{D}_{\textbf{g}}(\frac{\textbf{z}_0+\boldsymbol{\phi}(0,\textbf{z}_0,h,\textbf{g})}{2})$ where $g(\textbf{z}) = (v,f(y))^T$

How do I go about showing $\textbf{D}_{\textbf{g}}(\frac{\textbf{z}_0+\boldsymbol{\phi}(0,\textbf{z}_0,h,\textbf{g})}{2})$ = \begin{bmatrix} 0 & 1 \\ f'(\frac{y_0+y_1}{2}) & 0 \\ \...
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1answer
11 views

Show that $H(v, y) = \frac{v^2}{2} - F(y)$ is a first integral

$H(v, y) = \frac{v^2}{2} - F(y)$ is a hamiltonian of $y' = v, v' = f(y)$ Linear first integrals are of the form $I(x) = b^Tx + c$ where $b \in ℝ^d $ and $ c \in ℝ$ Quadratic first integrals are of ...
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0answers
22 views

Issue in applying partial chain rule

I was reading through a derivation and saw this I think it's using the partial derivative chain rule but I'm not sure how it's being applied in this case. Any ideas? We always used to do this little ...
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0answers
13 views

Leibniz rule generalization for partial derivatives

i'm searching for a way to comupute the m'th partial derivative of the product of n multivariable functions, namely i want to find some general leibniz formula to evaluate : $\prod_{i=1}^{m}$ $\frac{\...
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0answers
12 views

Generalized Leibniz rule for multivariable functions.

i want to know if there is some Leibniz formula to evaluate m'th partial derivative of the product of n multivariable functions, more precisely i want to evaluate : $$ \frac{\partial}{\partial x_{1}}...
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0answers
17 views

In the definition of partial derivative, why the function must be defined on an open set?

On the page Partial derivative on Wikipedia, the following formal definition was found: I am wondering if in this definition, the condition that $U$ being open is always necessary. For example, if in ...
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1answer
34 views

How to calculate the $\frac{\partial det(\mathbf X)}{\partial \mathbf X}$ and $\frac{\partial tr(\mathbf X^n)}{\partial \mathbf X}$

How to calculate the $\frac{\partial det(\mathbf X)}{\partial \mathbf X}$ and $\frac{\partial tr(\mathbf X^n)}{\partial \mathbf X}$ by using Frobenius product?i tried to begin the calculation,but i ...
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0answers
19 views

Is taking the partial with one variable held constant the same as the derivative of that function?

Consider $u(x, y).$ Let $v = \frac{\partial u}{\partial x}.$ Now let $w(x) = u(x, 0).$ My question is, is $v(x, 0) = \frac{d w}{d x}?$ I have a feeling this is true but I'm not quite sure...
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1answer
36 views

Matrix calculus (simple question)

Let $u_{0}$ be a row vector of real numbers, $U$ be a matrix of real numbers, $x$ some row vector. Then let $v = xU + u_{0} $ My question is how you would calculate $$ \frac{\partial v}{\partial u_{...
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1answer
28 views

Summability of double partial derivatives of $\frac{1}{|x|}$ in dimension $3$

I know the fact that its laplacian is equal to the Dirac delta function. However, is it true that its partial derivatives of order 2 belong to $L^{1}(\mathbb{R}^3)$? And how should I show that, in ...
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0answers
13 views

Finding the second order partial derivative of an arbitrary function.

I need to find the higher order partial derivative of a function. (Need to find: $\frac{\delta ^2 f}{\delta u^2}$ and $\frac{\delta ^2 f}{\delta v^2}$) The problem is that the function is not ...
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0answers
35 views

Epidemic spacial diffusion problem.. how to draw a graph of differential system of second order with partial derivative

I'm currently working on how to simulate the diffusion of an epidemic in a population. If we don't consider that the population is moving in space, then the differential system is the following: ...
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0answers
16 views

Cauchy-Riemann equations for $z=x+iy$ and $f(z)=R(x,y)e^{i\theta(x,y)}$

I know the following forms of the Cauchy-Riemann equations (please correct me if I made a mistake): $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$, $\frac{\partial u}{\partial y} = -...
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0answers
21 views

Existence of $f_{xy}$ if $f_{yx}$ is continuous

Is there a function $f\colon D\subset\mathbb{R}^2\rightarrow\mathbb{R},\ (x,y)\mapsto f(x,y)$, such that $\frac{\partial^2f}{\partial x\partial y}$ exists everywhere and is continuous, but $\frac{\...
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0answers
20 views

Finite difference replacement of a PDE

I'm having trouble with this question. I think that of of the partial derivatives should be looking for finite difference approximation of these two derivatives using the Taylor series expansion but ...
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0answers
32 views

Applying neumann boundary conditions to diffusion equation solution in python

For the diffusion equation $$\frac{\partial u(x,t)}{\partial t}=D\frac{\partial^2 u(x,t)}{\partial x^2}+Cu(x,t)$$ with the boundary conditions $u(−\frac{L}{2},t)=u(\frac{L}{2},t)=0$ I've programmed ...
2
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1answer
41 views

How to make it formally correct?

Can someone help me formalizing this statement: $$ z= x^0 +ix^1 $$ And therefore $$ \frac{\partial}{\partial z} = \frac{\partial}{\partial (x^0 +ix^1)} = \frac{\partial}{\partial x^0} + \frac{1}{i} \...
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1answer
30 views

How do I show that the system is hyperbolic if $u^2 + v^2 > c^2$

I know that for a system to be hyperbolic it must have 2 real distinct eigenvalues $\lambda$ where $det(B-\lambda A)=0$ my system of equations are: $(pu)_x + (pv)_y =0$ $p(uu_x + vu_y) + c(p)^2p_x =...
3
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4answers
76 views

Cancellation rules for partial derivatives

Is it possible to do some kind of simplifications on an expression like $$ f : x, y \to \mathbb{R} \\ \frac{\frac{\partial^2 f}{\partial y^2}}{\frac{\partial f}{\partial y}} = \frac{\partial f}{\...
2
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1answer
32 views

$u_{xx} + u_{xy} + u_{yy} = 0$ in canonical form

How do i put $u_{xx} + u_{xy} + u_{yy} = 0$ in canonical form? $a=1, b=1/2, c=1 $ implies that it is elliptic as $b^2 - ac <0$ $dy/dx = \lambda$ where $a\lambda^2-2b\lambda+c=0$ gives $\lambda = \...
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1answer
25 views

$d^2u/dx^2 - d^2u/dy^2 = f(x,y)$ $\rightarrow$ $d^2u/dξdη = \frac{1}{4}f(\frac{1}{2}(ξ+η),\frac{1}{2}(η-ξ))$

$$\frac{\partial^2u}{\partial x^2} - \frac{\partial^2u}{\partial y^2} = f(x,y) \implies \frac{\partial^2u}{\partial ξ\partial η} = \frac{1}{4}f\left(\frac{1}{2}(ξ+η),\frac{1}{2}(η-ξ)\right)$$ Setting ...
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1answer
10 views

How to find $\partial\chi^2/\partial b$ when $\chi^2=\sum_{i=1}^N\dfrac{D(x_i)-a-b(x_i)^2}{\sigma_i^2} $? [duplicate]

How do I find How to find $\partial\chi^2/\partial b$ when $\chi^2=\sum_{i=1}^N\dfrac{D(x_i)-a-b(x_i)^2}{\sigma_i^2} $? My attempt: \begin{align*} \dfrac{\partial}{\partial b}\sum_{i=1}^...