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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2
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2answers
26 views

Solving inhomogeneous PDEs when you can't separate variables

$$4+U_y - U_{xy} =0, \quad U(x,0)=0, \quad U_y(0,y)=3y^2 .$$ Usually I can solve these kind of problems with separation of variables, so I tried $$ U=XY, \quad U_y=XY', \quad U_{xy}=X'Y' $$ $$ ...
0
votes
2answers
41 views

Gradient vector proof

Question: Prove that a normal vector to the surface $f(x,y) = \sqrt {xy}$ at any point on the surface is perpendicular to the line joining the point to the origin. I am not sure how to do this. ...
1
vote
1answer
12 views

How to find the normal line of $z = 4x^2 + y^2 - 78$ at $(2,1,-61)$?

How do I find the normal line of $z = 4x^2 + y^2 - 78$ at $(2,1,-61)$? I have found that the tangent plane is $z-16x-2y=95$ but I don't know how to find the normal line. The answer is: $$\frac{2 ...
0
votes
1answer
41 views

Calculate $\frac{\partial}{\partial x_k}(\frac 1{|x-y|^{n-2}})$ and $\frac{\partial^2}{\partial x_j \partial x_k}(\frac 1{|x-y|^{n-2}})$?

I want to take first partial derivative w.r.t. $x_i$ (for $i,j,k=1,\ldots,n$) of $$x\mapsto\frac 1{|x-y|^{n-2}},\quad x\neq y.$$ where $y\in\mathbb{R}^n$ is fixed. Can I ask here if the following ...
2
votes
2answers
57 views

Solution of partial differential equation

Solve the differential equation, $$ z=\frac{\partial z}{\partial x}x + \frac{\partial z}{\partial y}y+ (\frac{\partial z}{\partial x})^2 + \frac{\partial z}{\partial x}\frac{\partial z}{\partial y}+ ...
0
votes
0answers
30 views

Problem finding the tangent plane and the normal line of an surface [on hold]

Good night, I have a serious problem when I try to find a tangent plane for the following surface at the point $P$: $$x^{2}+y^{2}+z^{2}=6, \hspace{4mm} P=(-1,-2,3).$$ I make this: $\nabla ...
0
votes
1answer
27 views

Problem solving a partial derivative with a integral. [on hold]

Good night, i have a serious problem solving this partial derivative: $f(x,y)=\int_{y}^{x}e^{t^{2}}dt$ I don't know how i can start this, please give me a help, don't do it the exercise, only explain ...
3
votes
1answer
31 views

How can I actually solve this kind of partial differential equations?

$$ x \frac{\partial z}{\partial x}+t \frac{\partial z}{\partial t}+y \frac{\partial z}{\partial y}=xyt$$ I can see that one soultion for this equation is $$z=(1/3)xyt+ C$$ however how can one solve ...
1
vote
1answer
44 views

Differentiation Involving Determinant.

I have to compute the following differentiation : $$\frac{\partial}{\partial\sigma^2}\det[\mathbf X_{p\times n}'(\sigma^2 \mathbf I_{n}+\mathbf Z_{n\times q}\mathbf G_{q\times q}\mathbf Z_{q\times ...
1
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0answers
8 views

How to express a homogenous function using an infinitely recursive matrix operation?

As is well known, if the multivariate function $f(\mathbf{x})$ is homogenous of degree $h$, then the partial derivatives of $f$ are homogenous of degree $h-1$. Also, say that we know $f$ is ...
1
vote
0answers
25 views

Is the derivative of a Bessel function really that complicated?

In this blog http://blog.wolfram.com/2016/05/16/new-derivatives-of-the-bessel-functions-have-been-discovered-with-the-help-of-the-wolfram-language/ they give this ridiculous complicated expression for ...
0
votes
0answers
18 views

Showing the continuity of $\partial_{x_ix_j}v$ and $\partial^2_tv$

How to show that if $\begin{cases}\partial^2_tv(v,t;s)-\Delta v(x,t;s)=0\quad \text{for}\ (x,t)\in\mathbb R^2\times\mathbb R_{>s}\\v(x,s;s)=0,\quad\partial_tv(v,s;s)=f(x,s)\quad \text{for}\ ...
1
vote
1answer
61 views

How many partial derivatives does a multivariate polynomial have?

My motivation for this question is from the following toy example; define the (nondeterministic) finite state machine generated by the polynomial $f(x_0 , ... , x_n) \in \mathbb{Z} [x_0 , x_1 , ... , ...
3
votes
1answer
24 views

Explicit Finite Difference Scheme For Approxating a p.d.e

$\frac{du}{dt} = \frac{d}{dx}[\frac{1}{x^2+1}\frac{du}{dx}]$ I am trying to approximate this pde with a finite difference scheme but I am confused with the d/dx. Do I just take the derivative of ...
2
votes
1answer
32 views

Derivative formalism question

After seeing a lot of integrals going like $$\int f(x,y) \,dxdy = \int f(x,y)\,dA$$ I am wondering wether it is allowed to write something like this: $$\frac{d f(x,y)}{dA} = \frac{\partial^2 ...
0
votes
2answers
42 views

Is the Hessian symetric for $z^3+y^2+xy+yz+3x-3z$?

I wish to study the function $f(x,y,z) = z^3+y^2+xy+yz+3x-3z$ and find its extreme values. I search for values for which $\nabla f(x,y,z) =0$. $$\frac{\partial f}{\partial x} = y+3$$ ...
1
vote
1answer
11 views

Sign of third order derivative

We have a concave function $f(x,y)$, i.e. the Hessian matrix has non-positive elements. Can we show $\dfrac{\partial^3f}{\partial x \, \partial y^2} \le 0$?
-3
votes
0answers
10 views

calculus of a derivative (stochastic calculus) [closed]

I have a doubt : I know that if $x_{t}=\int_{0}^{t}\gamma(s)dW_{s}$ (with $W_{s}$ a brownian motion), we have : $dx_{t}=\gamma(t)dW_{t}$ What about if $x_{t}=\int_{0}^{t}\gamma(s,t)dW_{s}$. Do I have ...
0
votes
1answer
48 views

The Laplacian in Polar Coordinates

When solving the Laplacian in polar coordinates, $x=r\cos\theta$ and $r^2=x^2+y^2$. When finding $\frac{\partial u}{\partial x}=\frac{\partial u}{\partial r}\frac{\partial r}{\partial ...
2
votes
1answer
37 views

Confusion with changing variables in second order DE

So in my physics assignment, we're given Schrodinger's equation: $$\frac{-\hbar^2}{2m}\frac{d^2\psi}{dx^2}+e\xi x\psi=E\psi$$ We're asked to substitute a function of the form $w(x)=Ax+B$ to arrive at ...
0
votes
0answers
14 views

Stochastic gradient descent in neural network with logistic activation function

I am trying to derive the update rules for a unit of a neural network. To simplify, let's assume that need to perform a binary classification task on a dataset $\mathbf{X} = \{\mathbf{x}_i\mid ...
2
votes
0answers
19 views

Parametrisation of a differential equation by arc length

Suppose I have the following PDE: $\frac{\partial y}{\partial t} = \frac{\partial}{\partial x}\left[(1-y)^3y^3\left(\sin \theta + \frac{\partial y}{\partial x}\cos \theta\right) \right]$ I notice ...
-3
votes
2answers
23 views

difference of partial derivative and its reciprocal [closed]

This is a rather simple question that I would like an explanation to. why is $$ \frac{\partial x}{\partial y}-\frac{\partial y}{\partial x}=0 $$ Similarly why is curl r = 0 where r is a vector ...
0
votes
2answers
32 views

Find the critical point of the following function

Let $f(x,y)=\dfrac{1}{2}x^2+cos(y)$ I want to find its critical points: $\dfrac{\delta f}{\delta x} = x$ $\dfrac{\delta f}{\delta y} =-sin(y)$ Now I have to solve the following system: $\left\{ ...
0
votes
2answers
22 views

utility function question from my textbook

Suppose there are two goods with prices $ p₁ = 2, p₂ = 5, $ the income is $ M = 40 $ and the utility function is $ U (x₁, x₂) = (x₁)^⅓ . (x₂)^ ½, $ Find the optimum consumption plan. Attempt: I do ...
-1
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0answers
15 views

economics partial derivative / consumer surplus question

Answer ALL parts of this question. (a) Explain what is meant by a „partial derivative‟, (you should include the definition of a partial derivative). For the function $z$ defined below, determine all ...
0
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0answers
15 views

When hessian is zero , second derivative test fail

I am not convinced that if the hessian is zero then the second derivative test fails .. I asked the question and proved it before but I found no answer prove if Hessian determinant is zero then ...
0
votes
0answers
28 views

How to calculate partial derivative of many values?

I have a function for example : $$f(x,y) = x^5 + 3xy + \cos(xy)$$ It's easy to calculate the partial derivative of $x$ or $y$. But how to calculate the partial derivative of $x$ AND $y$, $[ f'x,y ]$
-1
votes
2answers
54 views

Without Extreme Value Theorem, how do we find absolute extrema?

I have to find and classify the critical points of the following functions and then state which relative extrema are absolute extrema. $$f(x,y) = x^3 - y^3 - 2xy + 6$$ $$f(x,y) = xy + 2x - ...
0
votes
1answer
18 views

Jacobian matrix for change of variables from Cartesian coordinate system to Spherical (Geographic) coordinate system.

I am trying to obtain the Jacobian matrix for a change of variables from Cartesian coordinate to spherical coordinates. My spherical coordinate system is a conventional right-handed Geographic ...
0
votes
0answers
55 views

What does this notation mean? $∂df^2 / ∂x$

I am given variance $\sigma_x$ and function $y=f(x)$ According to my book, the following equation gives the new variance of $y=f(x)$. But I'm not sure what this notation means, as in what this ...
3
votes
4answers
60 views

Partial derivatives for polar coordiantes

I'm given that $\varphi = \arctan\left(\frac{y}{x}\right)$ and I'm asked to show that $$\frac{\partial x}{\partial \varphi}=-r\sin\varphi$$ I've tried to do this and I'm pretty sure this isn't true. ...
1
vote
0answers
12 views

Extremum and nonsymmetric Hessian matrix

If $F$ is assumed only to have all second partial derivatives (but we don;t assume that they are continuos) than it could happen that the Hessian matrix in some stacionary point is nonsymmetric. Is is ...
3
votes
1answer
26 views

Partial Derivative with product & chain rule

I cannot for the life of me work out the answer to this partial derivative. $$\frac{\delta}{\delta x}\left(\frac{x^2}{(x+y)^2(x+z)^2}\right) $$ My first thought was: Split into two equations: ...
1
vote
1answer
64 views

Finding stagnation points and stream function

Sorry for the lack of latex. The question I want to ask would need all this info and it would take very long to write it. (a) Irrotational flow means $\nabla \times \textbf u =0$ so we can define ...
1
vote
0answers
23 views

Find the particular solution given the auxiliary conditions

Let $z=f(x+cy)+g(x-cy)$ where $c$ is a constant and the functions $f$ and $g$ are twice differentiable. Deduce the solution of the equation subject to the auxiliary conditions $$z(x,0)=x \text{ and ...
2
votes
0answers
23 views

Perturbations to a matrix causing drastic changes to matrix inverse.

I'm reading this article about matrix norms because I want to understanding the math behind SVD. One of the interesting issues it brings up quite soon is the effect of perturbations to a matrix on ...
1
vote
0answers
36 views

prove if Hessian determinant is zero then second derivative test fails

I know if Hessian determinant is zero then second derivative test fails.. But I find that the following proof is a contradiction ! Moving along the unit vector $$u=u_1 i + u_2 j$$ and knowing that ...
0
votes
0answers
8 views

Can 'Frobenius product method' be used to get analytic expression for **vector derivative**?

this objective function is shown as follow: $$\min_{u*, i*}\sum_{ui}c_{ui}(p_{ui}-x_{u}^Ty_i)^2 + \lambda(\sum_u\|x_u\|^2 + \sum_i\|y_i\|^2) + \lambda_f(\|x_u-\frac{1}{|N(u)|}\sum_{f \in ...
4
votes
1answer
51 views

basic calc: partial derivative of $f(x,y) = x + y$

It has been many years since I had Calculus, and I am trying to brush up on some basics. I am told that if $f$ is defined as: $$f(x,y)=x+y$$ then: $$\dfrac{\partial f}{\partial x}=1$$ I expected ...
1
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0answers
36 views

Mixed partial derivatives-a counterexample

The well known theorem of Schwarz states that if $f \in C^k(U)$ then all partial derivatives up to $k$-th order of $f$ are equal. There is a well known example of function of class $C^2(R^2)$ such ...
0
votes
0answers
25 views

Derivative of kinky function

I have the following result: $$-\frac{1}{2}y^2(\frac{\partial^2f}{\partial x^2}-\frac{\partial f}{\partial x}) = -\frac{1}{2}y^2K\delta_{x=\log(K)}$$ where $$f(x) = (K-e^x) \mathbb{1}_{\{x \leq ...
2
votes
1answer
32 views

Relationship between two-equation constrained optimization and one-equation version

I am learning about the Lagrange multiplier. Here's what I understand so far. Suppose a point $P$ is a minimizer of $f(x)$ subject to $g(x)=0$. Then any movement along that level-curve of $g$ must ...
1
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0answers
19 views

Derivation Gaussian Mixture Models log-Likelihood

I'm trying to understand the derivation of the log-likelihood function for Gaussian Mixture Models. According to my records the following steps are made. The log-likelihood function is defined as: $ ...
1
vote
1answer
86 views

How to integrate a PDE?

How does one integrate this PDE with respect to $x$? $$u\frac{\partial u}{\partial t}+ u^2\frac{\partial u}{\partial x}+u\frac{\partial^3 u}{\partial x^3}=0$$ My idea is to rewrite this equation as ...
1
vote
1answer
43 views

Derivative with respect to a derivative

Let $q=q(t)\in C^1(\mathbb{R})$ and $V=V(x)\in C^1(\mathbb{R})$. My book uses the following fact over and over again $$\frac{\partial V(q)}{\partial \dot{q}}=0.$$ Why is this true?
1
vote
1answer
27 views

How to show that a function is differentiable even though its partial derivatives in origin don't exist

I have a function $ f(x,y) = \begin{cases} (x^2+y^2)\sin(\frac{1}{x^2+y^2}), & (x,y)\neq(0,0) \\ 0, & (x,y)=(0,0) \end{cases}$ and I need to show that $f(x,y)$ is differentiable, even though ...
1
vote
1answer
63 views

Elimination of λ and μ in Lagrange method of multipliers both constraints are nonzero

QUESTION: Determine maximum value of OP, O being origin of coordinates where P describes the curve $x^2 + y^2 +2z^2=5, x+2y+z=5$? Here using lagrange method of multipliers we have two constants λ and ...
0
votes
1answer
31 views

Please help me with this partial derivative

$z=f(u,v)$ ; $u=x^2 + y^2$ and $v=2xy $ FIND $(∂z/∂x)^2 + (∂z/∂y)^2$ Now, i am confused on how to differentiate z, (do i diffrentiate u and v both or what)...please help me with the procedure!.
1
vote
1answer
17 views

Chain Rule, Piecewise Derivative

I have a function $h(a,b)=g(f(a,b))$ where $f(a,b)$ is a smooth, continuous, multivariate function and $g(x)$ is a piecewise function s.t. $$g(x)=\begin{cases} 1, & 0 \leq x \leq 1 \\ ...