Articles of partial derivative

Question regarding partial derivatives 1

I’m trying to tackle the following question Let $f:\Bbb{R}^2\to\Bbb{R} \ , \ (x_0,y_0)\in\Bbb{R}^2 \ , \ \underline{u}=(u_1,u_2)\in \Bbb{R}^2$ where $\underline{u}$ is unit vector. Let $g(t)=f(x_0+u_1t,y_0+u_2t)$. Show that $D_{\underline{u}}f(x_0,y_0)=g'(0)$. My try: First, I think that it should be given that $g$ is differntiable at $x=0$. Now, $$D_{\underline{u}}f(x_0,y_0)= \lim_{t\to 0}\frac{f\left((x_0,y_0)+t\underline{u}\right)-f(x_0,y_0)}{t}=\lim_{t\to 0}\frac {f(x_0+u_1t,y_0+u_2t)-f(x_0,y_0)}{t} \\ g'(0)=\lim_{t\to 0}\frac{g(0+t)-g(0)}{t}=\lim_{t\to 0}\frac{f\left(x_0+u_1t,y_0+u_2t\right)-f(x_0,y_0)}{t}$$hence, the […]

Derivative of the Frobenius norm with respect to a vector

I am trying to calculate the derivative of an energy function with respect to a vector $\mathbf{x}$. The energy is given by: $\psi(\mathbf{F}) = \lVert\mathbf{F}-\mathbf{I}\rVert_F^2.$ Where $\mathbf{F}$ is a square matrix that is a function of $\mathbf{x}$ (a column vector): $\mathbf{F(x)} = (\mathbf{x}\cdot\mathbf{u^T})\mathbf{A}$ $\mathbf{u^T}$ is a constant row vector and $\mathbf{A}$ is a constant square […]

Prove partial derivatives exist, but not all directional derivatives exists.

During my analysis course my teacher explained the difference between partial derivatives and directional derivatives using the notion that a partial derivatives looks at the function as approaching a point along the axes (in case of of the plane), and a directional derivative as approaching a point from any direction in the plane. He also […]

If $F(x,y)=0$, prove $\frac{d^2y}{dx^2}=-\frac{F_{xx}F_y^2-2F_{xy}F_xF_y+F_{yy}F^2_x}{F_y^3}$

If $$F(x,y)=0$$ prove $$\frac{d^2y}{dx^2}=-\frac{F_{xx}F_y^2-2F_{xy}F_xF_y+F_{yy}F^2_x}{F_y^3}$$ I tried $$\frac{dy}{dx}=-\frac{F_x}{F_y}$$ Then $$\frac{d^2y}{dx^2}=-\frac{F_{xx}F_y-F_xF_{xy}}{F_y^2}$$ I do not know where I got wrong… any help? Thanks~

Computing second partial derivative with polar coordinates

Consider the polar coordinates with $x=r\cos\theta$ and $y=r\sin\theta$. I can show using the chain rule that $$\frac{\partial}{\partial x} = \frac{x}{r} \frac{\partial}{\partial r} -\frac{y}{r^2} \frac{\partial}{\partial \theta}$$ What is the method to compute $\dfrac{\partial^2}{\partial x^2}$? I don’t know how to do it.

Best method to solve this PDE

I need to solve this Partial Differential Equation for $\lambda(x,y)$, $$\frac{\partial \lambda}{\partial x} + h(x,y)\frac{\partial \lambda}{\partial y} – \lambda \frac{\partial h}{\partial y} = 0$$ where $$\frac{dy}{dx} = h(x,y).$$ The additional information given is $\lambda$ is a bivariate polynomial in $x$ and $y$. My initial approach was to try using the method of characteristics, but I […]

How to find the direction vector of a ball falling off an ellipsoid?

A tiny ball is placed in top of an ellipsoid $3x^2+2y^2+z^2=9$ at $(1,1,2)$. Find the three-dimensional vector $\underline u$ in whose direction the ball will start moving after the ball is released. I feel this problem involves usage of gradients but not sure how to tackle it. EDIT the solution shouldn’t use physics knowledge and […]

Partial Derivative v/s Total Derivative

I am bit confused regarding the geometrical/logical meaning of partial and total derivative. I have given my confusion with examples as follows Question Suppose we have a function $f(x,y)$ , then how do we write the limit method of representing $ \frac{\partial f(x,y)}{\partial x} \text{and} \frac{\mathrm{d} f(x,y)}{\mathrm{d} x}$ at (a,b)? What is the difference? Imagine […]

Example of non-differentiable continuous function with all partial derivatives well defined

Give an example of a function $f : \mathbb{R}^3 \to \mathbb{R}$ such that the partial derivatives exist at $(0,0,0)$, and $f$ is continuous at $(0,0,0)$, but it is not differentiable at $(0,0,0)$. Any hint?

Minimality in the case of partial derivatives and Sobolev spaces?

I am trying to understand this question here that considers Sobolev spaces apparently and hence partial derivatives. What is the definition of minimality there? Is the minimality defined by cardinality or by the less-or-equal operation i.e. “x is the minimal element if it is less or equal to any other element” where the less-or-equal can […]