Use this tag for questions about differential and integral calculus with more than one independent variable. Some related tags are (differential-geometry), (real-analysis), and (differential-equations).

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1answer
23 views

$1 \times 3$ tangent plane vs. gradient

Say we have some function $f(x,y,z)=xyz$ or whatever. It goes from $\mathbb{R}^3$ to $\mathbb{R}$. If we want the tangent plane of the function, we need to differentiate it, and this gives us a $1 ...
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0answers
21 views

How do you know when the derivative is a matrix or a sum?

For the derivative of a function, sometimes I see the derivative written as a matrix of the partial derivatives, and sometimes I see the derivative written as a sum of the partial derivatives. When do ...
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1answer
96 views

Max/min values attained by a function along a path

Say we have some arbitrary function $f(x,y) = xy$ or whatever. It doesn't have to be a scalar-valued function (though for my question it might be a restriction, not sure. Which is why I'm asking). If ...
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2answers
60 views

How to plot a surface in maple where the range is given by an expression, not constants?

Im trying to plot the surface $z=(1+x^2)/(1+y^2)$ , but specifically the part of the surface that is above $|x|+|y|\leq1$. Cant seem to find any information on how to produce a plot in maple, where ...
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1answer
26 views

Limits of $(e^{xy}-1)/y$

How would you solve this limit? It exists and is equal to 0 but I have no idea how to show it. $$\lim_{(x,y)\rightarrow(0,0)}\frac{e^{xy}-1}{y}$$
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0answers
33 views

Using the inverse function theorem to show that there is a “projection”

I'm trying to prove this result: Let $A\subset \mathbb{R}^n$ be an open set and $f:A\subset \mathbb{R}^n\to \mathbb{R}^m$, $m\leq n$, a function of class $C^{p}$. Let $x_0\in A$ and suppose ...
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2answers
267 views

Question regarding Stokes' theorem

I have a very simple question. The Calculus book I am using provides this question (along with a solution) as an exercise in Stokes' theorem. First of all, I have no idea why they have a picture of a ...
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0answers
163 views

Conversion of covariance matrix from Cartesian to Spherical coordinates for integration

I have to perform a convolution of a function in polar coordinates $\rho(\textbf{x}) = \rho(r,\theta,\phi)$ with a function $P(\textbf{x}) = P(x,y,z)$ in cartesian coordinates. $\int ...
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1answer
46 views

$f:R^2\to R$: Determining the Nature of a Critical Point when the Second Derivative Test Fails

I'm reviewing for a final exam tomorrow. This is an exercise that I am having trouble with: The function: $f(x,y)=x^2-y^4$ I determined that there is one critical point, at $(0,0)$. I determined ...
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2answers
85 views

Use Polar Coordinates to Find the Limit…

Use polar coordinates to find the limit. [If $(r, \theta)$ are polar coordinates of the point $(x, y)$ with $r \geq 0$, $r \to 0^+$ as $(x,y) \to (0,0)$)] $$\lim \limits_{(x,y) \to (0,0)} ...
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2answers
51 views

Extrema of two variable function

Find extrema of $f(x,y)=x^2-xy+y^2$ from set $M=\{ [x,y] \in \mathbb{R}^2;|x|+|y|\le1\}$ I am solving this kind of problems for the first time and I am not sure what I am doing, what I have got ...
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3answers
407 views

Find a possible equation for the linear function g(x,y) shown in the graph

Can someone please help me understand how to start this problem? I have posted this up before but have not received any help. I can obviously see that the gradient is 4, that the line passes ...
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2answers
67 views

Multivariable calculus - find total derivative

I want to find the total derivative of the function $f: \mathbb R^n \to \mathbb R^n$, $f(x)=\frac{x}{|x|}$ If I was to copy what the teacher taught, I should find the limit of $\lim_{t \to 0} ...
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2answers
73 views

Multivariable calculus - explain what the teacher did

The teacher gave this exercise: Find $D_f(a)$ when $f: \mathbb R^n \to \mathbb R$, $f(x)=<x,\xi>^2$ where $\xi \in \mathbb R^n$. What I did: I wrote it as $$f(x)= (\sum_{i=1}^{n}x_i ...
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2answers
221 views

Surface integral on unit sphere

I'm struggling to calculate the surface integral in this question Find the area of the portion of the sphere $$z=\sqrt{1-x^2-y^2}$$ Which lies between the planes $z=0$ and $z=1$ Now I know the ...
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0answers
32 views

zeros and poles of meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$

I'm a bit confused regarding the following example: it is stated that $$ \frac{x^3+y^3+z^3}{x^2yz}$$ is a meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$, and then one is asked to find poles ...
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2answers
61 views

Finding the limit of two variable function [closed]

$\lim_{(x,y)\to (0,0)}(xy/(x^2+y^2)^\frac{1}{2})$ What is the limit of this function?
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0answers
44 views

How to compute the Jacobian Matrix of the next system?

I have a little problem with notation and I do not know how to work it out. I have the next system $$ \dot x = A(x)x, $$ where $x \in \mathbb{R}^n$ and the square matrix $A(x)_{n\times n}$ has ...
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0answers
31 views

What is $Df$ in Multivariable Calculus?

I wasn't in class for a week of lecture because of medical problems. It seems to do with differentiation but I cannot find how to find $Df$ given $f$ (don't even know what Df stands for) Here is the ...
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93 views

Hadamard variational formula Evans chapter 6 problem 15

This is Evans' chapter 6 problem 15. Consider a family of smooth, bounded domains $U(\tau) \subset \mathbb{R}^{n}$ that depend smoothly upon the parameter $\tau \in \mathbb{R}$. As $\tau$ changes, ...
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2answers
113 views

What is meant by an open boundary when specifying boundary conditions of PDEs?

When speaking about boundary conditions of PDEs, one speaks about Dirichlet, Neumann or Cauchy boundary conditions specified over the boundary which can be closed or open. For example, we say that ...
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2answers
258 views

volume inside sphere but outside hyperboloid

I am trying to find the volume inside the sphere $x^2 + y^2 + z^2 = 9$, but outside the hyperboloid $x^2 + y^2 - z^2 = 1$. by using a triple integral. for some reason i just cant seem to come up the ...
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2answers
49 views

How to resolve this diferential equation $y^2 y^{\prime}=x^3$

I see it is non-linear, but not sure if that is important here. I got the solution for the homogeneous in this way: $$y^2 y^\prime=0 \rightarrow y^\prime=0 \rightarrow \frac{dy}{dx}=0\rightarrow ...
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1answer
45 views

Surface integrals help?

I'm having trouble understanding visually how a surface integral works/calculates. For a standard double integral, function $f(x,y)$ and a rectangular region $U=[a,b]\times[c,d]$ then the double ...
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0answers
35 views

Can you use the chain rule when only one partial derivative is continuous?

Say the partial derivative with respect to $x$ of some function $f(x,y,z)$ exists and/or continuous, but the other partial derivatives don't exist. Can I still apply the chain rule if I'm only ...
0
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1answer
81 views

Volume of a solid

Find the volume of the solid in $\Bbb R^3$, bounded by $$y = x^2\\x=y^2\\z=x+y+30\\z=0 $$ For me setting the integral is the issue!
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0answers
79 views

Taylor series expansion of a function with one independent and one dependent variable.

If $y=y(x)$ What will be the Taylor series expansion of $f(x+h, y(x+h))$ about $(x,y(x))$ upto first order partial derivatives of $f$. I think it will be , please help. ...
3
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1answer
34 views

Finding $\frac{\partial ^8 f}{\partial x^4\partial y^4}$

Given the function $f(x,y)=\frac{1}{1-xy}$ find the value of$\frac{\partial ^8 f}{\partial x^4\partial y^4}(0,0)$. First I developed the function into a taylor series using geometric series ...
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1answer
56 views

Evaluating a surface integral of a paraboloid

Calculate the average value of $(1+4z)^{3}$ on the surface of the paraboloid $z=x^{2}+y^{2}$,$x^{2}+y^{2} \leq 1$ I'm not sure on how to start this problem. I have already found the area of the ...
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1answer
132 views

Find the area of the indicated surface

Find the surface area of the part of the sphere $x^2 + y^2 + z^2 = a^2$ inside the circular cylinder $x^2 + y^2 = ay$ ($r = a\sin(\theta)$ in polar coordinates), with $a > 0$. First time posting ...
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1answer
79 views

Triple integral in cylindrical coordinates question

Can someone explain the answer given split the integral into a cylinder and volume below a sphere? Thanks.
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1answer
139 views

Derivative of sum of two functions is the sum of their derivatives.

Suppose $x_0 \in U \subseteq \mathbb{R}^d$, $U$ open, and $f,g : U \to \mathbb{R}^m$ differentiable at $x_0$, then $$D_{f + g} (x_0) = D_f(x_0) + D_g(x_0).$$ MY ATTEMPT Put $r(x) = f(x) + g(x) ...
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1answer
329 views

Application of Implicit Function Theorem

So I'm a set of practice problems regarding this but I don't quite understand how to think about this... Example of a problem: $x^3 (y^3 +z^3 )=0$ and $(x-y)^3 -z^2 -7=0$ A point lies on the ...
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0answers
13 views

Prove that the Laplacian of the integral of a certain function is $0$

Let $f(x)$ be a continuous function. Define $$g(x,y)=\int_a^b\frac{yf(t)}{(x-t)^2+y^2}dt$$ Show that $\nabla^2g=0$
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1answer
27 views

Rewriting line integral for complex-valued function

Context: Suppose $f = \phi + i\psi$ is continuous and $\gamma(t):[a, b] \to \mathbb{C}$ is a curve. Then we define the integral of $f$ along $\gamma$ to be $$ \int_\gamma\!f = \int_a^b ...
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0answers
49 views

Taylor series like polynomials

Let $U$ be an open subset of $R^n$ and $f:U\rightarrow \mathbb{R}$ a function and $x\in U$ such that in a small neighbourhood of $x$ and for $\epsilon \in \mathbb{R^b}$ sufficiently small we have the ...
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0answers
78 views

Calculating the surface area with parabolic coordinates

Calculate the surface area of $x^2 + y^2 -18z=81$ with the surface $-4.5<z<0$ using this variable change: $$x=uv\cos\phi\\y=uv\sin\phi\\z={1\over2}(u^2-v^2)\\u,v\ge0,\ ...
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2answers
58 views

How could I solve this double integral question

Evaluate$$ \iint_R \left ( e^{-x-y} \right )dxdy, $$ where $R$ is the region in the first quadrant in which $x+y\leq 1$. I think the first step is $$\int_{0}^{1}\int_{0}^{1-y}\left ( e^{-x-y} ...
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2answers
76 views

Prove that the limit does not exist

I have to prove that the limit does not exist for the following function: $$\lim_{(x,y)\to (2,1)}\frac{4y^2-x^2}{(x-2y)^3}$$ I tried taking different paths for the limit, but I couldn't get anywhere ...
1
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1answer
52 views

Understanding Conservative and Curl

There are several things I need to clarify on Curl. 1) Is the conservativeness of a gradient field only applicable for a Closed curve? If the field is gradient and if c (curve) is not closed then ...
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0answers
137 views

Showing some complicated integral expression is bounded

In my research, I come across some expression I need to bound. I wish to show that the following integral is bounded in $t$ and $x$, i.e. the following supremum is finite: $$\sup_{t,x\in ...
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1answer
40 views

Integral inequality in $\Bbb R^n$

I came across this problem : Let $f\colon [a,b]\rightarrow \mathbb{R}^n$ a continuous vector valued function. Then it is true that: $$\left\Vert\int \limits_a ^b f(t) dt\right\Vert \leq \int ...
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1answer
43 views

Maximization problem in multivariable calculus

Let $f(x_1,\ldots,x_n) = x_1x_2\cdot \cdot \cdot x_n$. Let $A : = \{ x \in \mathbb{R}^n : x_1 + x_2 + \cdots + x_n = n , \; \; x_i \geq 0 \; \; \forall i \} $. I want to find the global max of $f$ ...
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1answer
164 views

continuity and differentiability of function of two variables

Let $f(x,y)$ be $$f(x,y): \begin{cases} x & \text{for } y = 0\\ x-y^3\sin\left(\frac{1}{y}\right)& \text{for } y \neq 0\end{cases} $$ then check continuity and differentiability at $(0,0)$. ...
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1answer
146 views

Differentiate vector norm by matrix

I've been trying to perform the following differentiation of a neural network: $$\frac{\delta||h(XW)\alpha-y||^2}{\delta W} = \frac{\delta}{\delta W}\sum_i(h(XW)_i\alpha-y_i)^2$$ Where $X$ and $W$ ...
3
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1answer
64 views

Differentiable function in n dimensions

If a function $f:\mathbb R^{n} \rightarrow \mathbb R^m $ is differentiable at a point $a$ can we say that there is a neighbourhood of $a$ such that $f$ is locally Lipschitz? (i.e. there is some ...
2
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1answer
36 views

Is it true that any norm could be taken to be the definition of differentiation on $\mathbb{R}^n$?

Let $||\cdot||$ be the 2-norm. Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be a function. Then we define $f'(x)=A$ if $0=\lim_{h\to 0} \frac{||f(x+h)-f(x)-Ah||}{||h||}$. If i take a different norm ...
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2answers
704 views

Find a linear function whose graph is the plane that intersects the xy plane along the line.

I have an exam in an hour and this one question is killing me! We enter our answers online so maybe I am just entering it wrong, but I can't for the life of me seem to get it right.. any help? Find a ...
0
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1answer
66 views

Check my answer - Differential of $P(A)=\det(A^{-1}-A)$

We are asked to find the differential of $P: GL_n(\mathbb R) \to \mathbb R$, $P(A)=\det(A^{-1}-A)$ and show it is differentiable. If we define $f(A)=\det(A)$ and $g(A)=A^{-1}-A$ then it is clear ...
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1answer
59 views

Find a plane that passes through the given points and is tanget to the graph

I have no idea about how to solve this. I know how to find a tanget plane to a surface, but I'm not sure if I understand what "passes through the points" mean. What is asked: Find a plane that passes ...