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
30 views

Values that make f(x,y) continuous

If I have the function $f(x,y)=|x|^a|y|^b$. How can I find the values of $a$ and $b$ that make the function continuous/discontinuous at (0,0).
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0answers
4 views

Multivariate Regression

Suppose there are $n$ variables that map through a function to a single output variable $r$. Given a set of 50-100 data sets with accepted input and output values that satisfy this relation, is it ...
4
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0answers
31 views

How do I do this change of variables?

Use a change of variables to evaluate: $$\iiint\limits_{D}xy\,\mathrm{d}V$$$D$ is bounded by the planes $y-x=0$, $y-x = 2$, $z-y = 0$, $z-y = 1$, $z=0$, $z=3$. I set $$u = y-x$$ $$v = z-y$$ $$w ...
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1answer
20 views

Can we do Taylor approximation in one direction

Let $f:\mathbb{R}^2\to\mathbb{R}$. Can we do Taylor approximation for only one variable $$f(x,y) \approx f(x_0,y) + \frac{\partial }{\partial x}f(x_0,y)(x-x_0) + \frac{1}{2}\frac{\partial^2}{\partial ...
0
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0answers
32 views

Proof of taking derivatives of both equal sides

I am curious about the proof of the following or whether the statement is true in general Assume that I have the following property: $f(x,y)=g(x,y,z)$ Can I assert that $D_xf=D_xg$ at any point ...
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0answers
14 views

Setting up a volume-finding calculation

I'm asked to find the volume inside the sphere $x^2+y^2+z^2=25$ and outside the cylinder $x^2+y^2=1$. I approached the volume $V$ in the following way: ...
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0answers
19 views

Implicit function theoroem [on hold]

Let $F = F(x, y, z)$ be a $C^1$ function which can be solved for any of the three variables in terms of the other two. Show that then $$ \frac{dx\,dy\,dz}{dy\,dz\,dx}=-1$$
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1answer
11 views

Non-singular derivative definition

I have a basic definition question. I am studying inverse function theorem, and I am stuck with what it means to say that for a $f'$ is non-singular? I looked it up in the internet, but it did not ...
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0answers
15 views

Partial Derivatives of a given function in $C^1$ [on hold]

Let $f:\Bbb R^3 \to \Bbb R$ be a given function in $C^1$. Find $d_xw$ and $d_yw$ in terms of $d_1f,\ d_2f,$ and $d_3f$ if $$w(x,y)=f(e^{x-3y},\ \sqrt{1+y^2},\ \ln(1+y^6))$$
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0answers
16 views

how to show the concavity of a very complex function?

To prove that S(n) have local maximum, I am thinking of taking second-order derivatives to $n$, then discussing the other parameter values. But S(n) seems way too complicated to me, I was wondering if ...
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0answers
15 views

Triple integral containing definite integral and exponentials with trigonometric functions

I am attempting to solve the following integral analytically: $$ \int_{z=5i}^{z=1} \int_{t=\csc^{-12}(z)}^{t=2} \int_{\theta=\sin^{t}(z)}^{\theta=t^2} {[\mathrm{e}^{t\cos(\mathrm{e}^{i \theta})} + ...
1
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1answer
35 views

Equivalent form of a double integral.

I am looking at the second question of this problem set: The iterated integral $\int_0^1 \int_{y/2}^1 e^{x^2} dx \, dy$ can be expressed as (a) $\int_0^1 \int_0^{2x} e^{x^2} dy \, dx$ ...
0
votes
1answer
12 views

Relation between minimum of a function and minimum of the sum of the same function and a linear term

I'd like to know if it's true that if given a function $f(x):X \mapsto \mathbb{R}$ and a vector $c \in X$, then if $$v = \arg\min_x f(x) + x^tc$$ one can say that $$v-c = \arg\min_x f(x)$$ Does this ...
0
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0answers
7 views

Looking for an alternative solution for optimal control problem

Let's say we have the following function ; $\intop_{0}^{\infty}\int_{0}^{N}V\left(C(t,\tau\right)dtd\tau$ and we want to maximise it according to the following constraint ; ...
1
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1answer
38 views

Prove whether a particular function is concave

Given the following equation: $$V(w) = - \frac{\alpha}{2} \left[ y_1(w) + y_2(w) + \int _{-\infty}^{+\infty} \vert y_1(w) - y_2(w) - x\vert f_{T1}(x)dx\right] \\- \beta \int _{w - y_1(w)} ^{+\infty} ...
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1answer
33 views

Evluating triple integrals via Spherical coordinates

Use Spherical coordinates to evaluate the triple integral $$\iiint_{\mathrm{x^2+y^2+z^2<z}}\sqrt{x^{2}+y^{2}+z^{2}}\, dV,$$ What I tried Converting $x^2+y^2+z^2<z$ to Spherical coordinates ...
2
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4answers
125 views

Find absolute maximum and minimum with domain

Find absolute maximum and minimum of the function $f(x,y)=3-x^2+y^2$ on the region $R = \{(x,y):1≥x≥0, 2≥y≥0\}$ I found that the gradient is $∇f(x,y)=(2x,2y)$ and that the critical point inside ...
0
votes
1answer
24 views

Differentiating both sides of an equality with respect to first variables? (Not answered)

I am proving a statement and the truth of the following proposition would help me with it. If anyone could say whether the proposition is true and give a hint how to prove it I would be very much ...
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0answers
37 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Proof:$\forall a,b;c,d\in\mathbb{R},a<b,c<d.$ $f (x+y) $ is ...
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2answers
41 views

How do I approach this double integral?

Let $R$ be the region inside $$x^2+y^2 = 1$$ but outside $$x^2+y^2 = 2y$$ with $x \ge 0 $ and $y \ge 0$ Let $$u = x^2 + y^2$$ and $$v = x^2+ y^2 - 2y$$ Compute $ \iint_R xe^y dxdy$ using this change ...
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0answers
24 views

2 variable limit

So, I understand why these bigger limit above does not exist (I'll name it 1), but I can understand why the other (2) is $0$. It seems to me that the $y^4/(x^6+y^8)$ is a non limited function and so ...
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0answers
17 views

Integration by parts partial derivatives

Given $$\int_x \int_t \Big( \frac{\partial}{\partial t}u(x,t) + \frac{\partial}{\partial x}f(u(x,t)) \Big) \phi(x,t)~~ dt dx = 0$$ How can I apply integration by parts in order to have the ...
1
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1answer
17 views

Rewriting triple integrals

I'm having trouble rewriting a triple integral. The question is rewrite the following integral in five different ways: $\int_0^1\int_y^1\int_0^z f(x,y,z) dx dz dy$ I am having trouble with ...
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2answers
49 views

Integration of $F(\sum_k x_k)$ over positive orthant

Problem Suppose we some function $F\left(\sum\limits_{k=1}^n x_k\right)$ over the positive orthant $[0,\infty)^n$. Show that this this is proportional to the integral $\int\limits_0^\infty ...
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0answers
9 views

Level curve of function with three variables

Sketch some level curves of the function f(x,y,z) = xy + yz, c=0 So when I set x=y=0, then z than can be any real number. Setting x=z=0, y can be any real number, same with x as well. Is this ...
2
votes
1answer
46 views

Compute a multiple integral$\iint_{[0,1]^2} (xy)^{xy} dxdy$

$$\text{Compute} :\iint_{[0,1]^2} (xy)^{xy} dxdy$$ I am thinking about changing the variable, $x=u,y={v \over u}$.But it doesn't work. I just found that the answer is$\int_0^1 t^t dt$.Maybe my idea ...
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2answers
15 views

Converting plane equation from $ax+by+cz=d$ to $r=a+\lambda b+\mu c$

The equation of the plane Π is $$2x + 3y + 4z= 48$$ Obtain a vector equation of Π in the form $r = a + λb + μc$, where a, b and c are of the form pi, qi + rj and si + tk respectively, and ...
1
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1answer
29 views

Change of variable in double integrals

I need help to solve the following question(s). a) Evaulate the integral $$\iint_D (x-y) \, dx \, dy,$$ where $D$ is the triangle with vertices $(0,0)$, $(-1,1)$ och $(4,2)$. b) Evaulate the ...
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0answers
35 views

How do I do this double integral (change of variable)

$B$ is the region bounded by $xy = 1$, $xy = 3$, $x^2 - y^2 = 1$, $x^2 - y^2 = 4$ Find $$\iint\limits_{B}x^2 + y^2 \,dx\,dy$$ using the change of variables: $$u = x^2 - y^2$$ $$v = xy$$ So I think ...
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0answers
26 views

What is the variance E[A]^2, statistics? [on hold]

$x(t)= A_i$, for $i \leq t < i + 1$ and $\{i = 0, 1 ,2 ,3,.....\}$. $A_i$ are independent variables, pmf of $P(A_i = 1) = P(A_i = -1) = 1/2$. Find the variance $E[A]^2$. I am so stuck on this ...
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1answer
18 views

Intersection of Level Curves and a Ellipse at a given angle

I am preparing for an exam and I'm going over previous administered tests. I have come across the following problem and have little idea how to tackle it. It goes as follows: Let ...
2
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1answer
46 views

Inverting a function

I am stuck with the following problem I am supposed to find the inverse of the function $g$ with $2$ variables, where $$\begin{align*}g&: R^2\to R^2 \\ g&(x,y)=(2ye^{2x}, xe^y)\end{align*}$$ ...
0
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1answer
35 views

How to find the normal vector of $xyz=1$

How do I find the normal vector of $ xyz=1 $ at $(a, b, c)$? Is the answer below correct? Because some answers on here are saying that the normal vector is $$ \Delta F = (f_x,\:f_y,\:-1) $$ So ...
2
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0answers
29 views

Directional derivatives with given values.

At the point (1,2), the function f(x,y) has a derivative of 2 in the direction toward (2,2) and a derivative of -2 in the direction toward (1,1). Find f_x(1,2) and f_y(1,2). Find the derivative of f ...
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0answers
9 views

Divergence using suffix notation

I am trying to show the divergence of $v(r) = ∇r^n$ where $r =$ ||r|| using suffix notation. The solution I am given says: $∇ · v(r) = ∂^2_ir^n = n∂_i(r^{(n−2)}r^i) = n((n − 2)r^{n−4}r^2+i + ...
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0answers
9 views

general inverse of multivariable function

Given $m<n$. Let $f:N\subset\mathbb R^m\mapsto \mathbb R^n$ be a differentiable function. I am looking for the condition(s) such that we can find a function $g:im(f(N))\mapsto \mathbb R^m$ ...
0
votes
1answer
18 views

Determining when $f(x,y) = x^{4/3} \sin(y/x)$ ($x \ne 0$) is differentiable.

Let $f$ be defined as follows: $$ f(x,y) = \left\{ \begin{array}{lr} x^{4/3} \sin(y/x) & \mathrm{if\ } x \ne 0 \\ 0 & \mathrm{if\ } x = 0. \end{array} \right. $$ I am asked to determine where ...
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1answer
15 views

Intermediate Integration Question

I'm having difficulty understanding why $$\int \left[ \left(\frac{dy}{dx}\right) ^2 + \left( y \right) \left( \frac{d^2 y}{dx^2} \right) \right]dx = \left( y \right) \left( \frac{dy}{dx} \right)$$
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0answers
7 views

A question from calculus my test(Curl, guess theorem )

the value of the integral $$ \iint rotF*n*ds \quad where \quad s-> x^2+y^2+z^2=4 \quad $$ and the normal is making a blunt angle with the Z axis, and $$ f=(zsinx-2y+1)i+(3x)j+(4xz+z^3)k $$ im ...
0
votes
1answer
27 views

Splitting partial derivatives

How come $$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial f}{\partial v} \frac{\partial v}{\partial x}$$ when $$ u = x\; cos \theta - ...
0
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1answer
15 views

Finding extrema of functions with 2 variables. [on hold]

Find the largest sphere inscribed inside tetrahedron witn vertices (1,0,0), (0,1/2,0), (0,0,1), and (0,0,0). Round the radius to 2 decimals.
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2answers
25 views

Showing that a multivariable function is one to one

I am stuck with the following problem I am given the function $f$ such that $f(x,y)=(x^2-y^2,2xy)$ I am supposed to show that the function is one to one. For a function to be one to one, $f'>0$. ...
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2answers
163 views

Can a region always be parametrized by a single function?

Can some connected region in $\Bbb R^n$, possibly with some other nice conditions on the region, always be parametrized by a single function $\vec r(x_1, x_2, \dots, x_k)$ (even if it may be easier to ...
0
votes
1answer
17 views

Gradient of an implicitely defined function?

For some function $F(x,y,z) = 0$, is the gradient $\nabla F $ always equal to zero? If you take the partial derivatives of both sides, you get zero for all of them. My book says: Which implies that ...
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0answers
8 views

Lagrange Multipliers

The Question: Find the minimum distance between the origin and the surface $x^2y -z^2 +9 = 0$. I've been able to find the critical points when $x =0$ and when x is not equal to zero but lamda is ...
0
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1answer
21 views

Is this function of 2 variables differentiable?

$f(x,y) = \frac{\sin(x^4+y^4)}{x^2+y^2}$ when $(x,y) \neq (0,0)$ and $0$ when $(x,y) = (0,0)$ Is f differentiable?
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vote
2answers
21 views

Chain rule for multiple variables?

What I've tried so far: $$F(x,y,z(x,y)) = 0$$ $$\implies \frac{\partial F}{\partial x} = 0$$ By the chain rule: $$\frac{\partial F}{\partial x} = \frac{\partial F}{\partial z}\frac{\partial ...
1
vote
2answers
36 views

Contradiction when differentiating?

Consider the function $F = x+y$. Let $x = t$ and $y= \cos t$. By directly differentiating, $$\frac{\partial F}{\partial x} = 1$$ and $$\frac{\partial F}{\partial y} = 1$$ Using the chain rule ...
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0answers
25 views

Question about a proof in Lang's undergraduate analysis

This is from page 580 of Lang's undergraduate analysis (2nd edition). I have difficulty in understanding the proof, hope that someone here can enlighten me. My questions are: i) On line 5 of the ...
0
votes
1answer
21 views

Geometric interpretation of derivative?

For some function $F(x,y) = 0$, $$\frac{dy}{dx} = \frac{-F_x}{F_y}$$ Can someone give me a geometric interpretatio of this? ($F_x$ and $F_y$ are the partial derivatives)