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Questions tagged [multivariable-calculus]

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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13 views

Change of variable in double integrals.

Problem Express the double integral $\iint_S f(x,y) \,dx \,dy$ as an iterated integral in polar coordinates,where $S$={$ (x,y)|x^2 \leq y,-1 \leq x \leq 1$}. Doubt I have difficulty finding ...
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1answer
19 views

Show that the product of the Jacobian and the inverse Jacobian is 1

I have seen the following fact in a textbook, but am having trouble proving it. If the Jacobian ("stretch factor" for change-of-variables) is given by $\left | \frac{\partial (x,y)}{\partial (u,v)} \...
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9 views

Conditions to tell that the total flux in a vector field across a closed surface is zero

When is the total flux across a closed surface zero. I am trying to find a set of values for an equation to prove that it has a total flux of 0 across a closed surface. I am not given any specific ...
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21 views

Points of Confusion About Second-Order Taylor Formula of Taylor's Theorem For Many Variables

My textbook has written the following for the second-order Taylor formula of Taylor's theorem for many variables: $$f(\mathbf{x}_0 + \mathbf{h}) = f(\mathbf{x_0}) + \sum_{i = 1}^n h_i \dfrac{\...
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Partial derivatives of a homogenous function

See demo of the theorem here : https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/hom/t) This is actually a follow up question on this one : Help to understand the proof of partial ...
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13 views

Regarding moving the differential inside of a double integral.

Forgive me it has been a few years since I took multivariable calculus so I am unsure about something. Say I am given something like $\int_x^y \int_x^y h(a,b)f(a,b|x,y)dadb = g(x,y)$. Specific ...
5
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1answer
60 views

Evaluate $\int\int\int_{[0,1]^3}\frac{dxdydz}{(1+x^2+y^2+z^2)^2}$

As in the title I have to evaluate this triple integral. $$\int\int\int_{[0,1]^3}\frac{dxdydz}{(1+x^2+y^2+z^2)^2}$$ I've been trying to solve this since a week ago. The first thing I've done was ...
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1answer
12 views

integrate over one cordinate like partial derivative

Probably my question is related to this question, but this question does not provide the answer to my question. Let $f:X\times Y \rightarrow \mathbb{R}$. Suppose I am interested in integration of $f(...
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27 views

prove laplacian equal zero [on hold]

how I can find the first and second derevative respect to x?
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2answers
51 views

Proof inequality $\frac{\sqrt{\pi}}{2}\sqrt{1-e^{-a^2}} < \int_0^a e^{-x^2}dx < \frac{\sqrt{\pi}}{2}\sqrt{1-e^{-2a^2}}$

I'm asked to prove the inequality $$\frac{\sqrt{\pi}}{2}\sqrt{1-e^{-a^2}} < \int_0^a e^{-x^2}dx < \frac{\sqrt{\pi}}{2}\sqrt{1-e^{-2a^2}}$$ After playing around for a while I was able to find (...
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1answer
29 views

a question about gradient

Let $f(x,y)=\ln\|\mathbf r\|$ where $\mathbf r=x\mathbf i+y\mathbf j$. Show that $\nabla f=\frac{\mathbf r}{\|\mathbf r\|^2}$. I attempt to calculate $\nabla\mathbf r$. But I have no idea how to ...
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2answers
40 views

How to find an area between bounds with polar coordinates? (Double integral)

The problem is: Find the area of the region that is bounded by x = 2 on the right and y = x/ √ 3  above, and by the circle $x^2$ + $y^2$ = x I have a general idea of how to find the ...
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2answers
58 views

Differentials in Multivariable Calculus

Does the idea of composing/decomposing the fraction notation of the derivative from/into differentials apply in multivariable calculus? I realize that this practice is considered non-standard and ...
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1answer
31 views

$\iint (x+y)e^{xy} dA$ . where A is region enclosed by $x-y = 1$, $x-y=4$, $xy = 1$, $xy=2$

Find the integral $\iint (x+y)e^{xy} dA$ . where A is region enclosed by $x-y = 1$, $x-y=4$, $xy = 1$, $xy=2$ So I tried the following substitution: $$u = xy$$ $$v = x-y$$ And want to find the ...
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1answer
44 views

Is the local max point also a global max?

Consider the following function $f$ of three variables, defined on $\mathbb{R^{3}}$: $$f(x,y,z) = 15x + xy - 4x^{2} - 2y^{2} - z^{2} + 2yz + 7$$ Is each of the critical points a local maximum? Is ...
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2answers
29 views

Temperature in a point is given by $T=xyz$. If you are in the point $(1,1,1)$, in which direction you should go to keep the same temperature?

I've found the following exercise: Suppose that the temperature in a point is given by $T=xyz$. If you are in the point $(1,1,1)$, in which direction you should go to keep the same temperature? I ...
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25 views

Existence of an isolated point such that $\frac{\partial u}{\partial y} = 0$.

Let $u$ be a smooth real valued function on $\mathbb{R}^2$. Is it possible for the derivative $\frac{\partial u}{\partial y}$ to have an isolated zero? I think the answer is no. Here is my reasoning: ...
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1answer
33 views

Is $z = x^2y^3(1-x-y)$ convex or concave?

Is there some kind of trick to defining the domain of the concavity/convexity (if it exists)? I have no idea how to work with the resultant hessian
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1answer
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Not sure about 2nd half of this question

I get the first part that you need the partial derivate of x in respect to y and y in respect to x to be equal for it to 'maybe' be a gradient. After this however I don't see what im trying to do or ...
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How to find the Volume of Dubai's Museum of the Future (An ellipsoid with an ellipsoidal hole in it)?

I'm an IB Mathematics HL student doing a math research report for school, I need to be able to model Dubai's Museum of the Future building. Do you know how I could model this shape mathematically and ...
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1answer
43 views

Finding an analytical solution to a simple 2D Finite Element Method problem

diagram Is it possible to find the value of scalar function u(x, y) anywhere in the region $\Omega$, given the following: $\nabla \cdot$ ($\nabla$u) = f, u(x,y) = g when y = 0, L$_2$ ($\nabla$u)$...
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0answers
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Derivative with single integral of variable limits, but with separate integration variable.

Say I have the following: $$f(t) = \int_0^t{e^{s^3\sin(t)}ds}$$ How do I compute $f'(t)$? My first thought was to use FTC, but the $s$ variable makes this confusing. I'm not completely sure where ...
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1answer
30 views

How to show whether limit of this function exists or does not exist?

The function is $$f(x,y)=\frac{2x}{x^2+x+y^2}$$ I want to check whether $$\lim_{(x,y)\to(0,0)} f(x,y)$$ exists or does not exist. I tried in polar coordinates and get the function $$f(r, \theta)=\frac{...
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0answers
29 views

Prove that it is possible to isolate $z$ as a function of $(x,y)$ in $x^2 z^2-z^3 y x=0$ near the point $(1,1,1)$

Prove that it is possible to isolate $z$ as a function of $(x,y)$ in $x^2 z^2-z^3 y x=0$ near the point $(1,1,1)$, but not near the origin. I just need the idea for start, cause its supposed to be a ...
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0answers
23 views

Validity of axes renaming

The position vector is $\vec{r}=(x,y,z)$. $\vec{A}$ is a vector field, $\vec{A}=(a_1,a_2,a_3)$ and $C$ is a constant. We have the function $$f=f(a_1x,a_2x,a_3x,Cx^2).$$ Can one then say, that ...
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2answers
22 views

Existence of a double integral of given function

Problem Let f be defined on the rectangle Q=[0,1]×[0,1] . f(x,y) is 1 when x=y and 0 elsewhere. Prove the double integral exist and equal to zero. Double I have no idea how to prove the existence ...
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1answer
30 views

Relation between Riemann sums and oscillation of a bounded function

Let $I$ denote a closed (hyper)rectangle in $R^n$ and $f:I \rightarrow \mathbb{R}$ be a bounded function. We use the following notations. (1) $\nu(f;a)$ is the oscillation of the function $f$ at a ...
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1answer
117 views

Finding the volume of a pseudosphere that has been parametrised in $\theta$ and $t$

I've got a problem calculating the volume of the top half of a pseudosphere. The pseudosphere is parametrised by $$\Phi(t,\theta) = \Big ( \frac{\cos(\theta)}{\cosh(t)}, \frac{\sin(\theta)}{\...
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1answer
11 views

Show equivalence of minimization techniques using Lagrangian multiplier.

Define a cost function : $$L(x,w) = \frac{1}{2} \sum_{i=1}^N\left(y^{(i)} - w^T\phi\left(x^{(i)}\right)\right)^2 + \frac{\lambda}{2}\sum_{j=1}^M w_j^2$$ Show that minimizing $L$ is equivalent to ...
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1answer
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Showing how the Jacobian connects volumes for change of coordinates

I'm having trouble understanding how exactly the Jacobian is transforming volumes. To understand this in more detail... if I have a two-dimensional integral and I'm trying to change the integration ...
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43 views

Partitions of unity of a compact subset of a smooth manifold

Let $M$ be a smooth manifold (without boundary if you want). Let $K$ be a compact subset of $M$. Let $\{(U_\alpha,\phi_\alpha)\}_{\alpha=1}^s$ be a finite number of charts of $M$ such that $\{U_\alpha\...
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A function $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ that maps a circle to a circle

Suppose f is a continuous function from $\mathbb{R}^2$ to $\mathbb{R}^2$ that maps a circle to a circle. How do I prove that f is differentiable? Will the function still be differentiable if ...
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Linear approximation of $3$ variable function and its maximum error?

I've got a problem which asks me to find linear approximation of multi-variable function and its maximum error. Here's the problem : By about how much will $$g(x,y,z)=x+x\cos(z)-y\sin(z)+y$$ ...
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1answer
19 views

Find limits along different paths of the form $y = mx$ [on hold]

I need to show that the limit: $\lim\limits_{(x,y)\to(0,0)}f_{y}(x,y)$ does not exist along different paths of the form $y = mx$, for $m$ is a real-number. $$f_y(x,y) = \frac{-3y}{\sqrt{x^2 + 3y^2}...
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1answer
41 views

Poincare type inequalities

I want to prove if following inequality holds: $$\int_0^1(f')^2\ dx\geq f^2(1)-f^2(0)$$ where $f$ is a function in $H^1([0,1])$ satisfying $\int_0^1f \ dx=0$. It is actually a one dimensional case of ...
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2answers
26 views

If f(x,mx) is algebraically reduced to contain only 'm', does it prove the limit does not exist?

Suppose you trace the path y=mx of a multivariable function f(x,y) to find the limit as (x,y)->(0,0). If f(x,mx) is algebraically reduced to contain only 'm', does it prove the limit does not exist? ...
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19 views

Derivative with respect to Vector: Notation

Let $x$ and $y$ be vectors in $\mathbb{R}^n$. First, does the above notation imply that $x$ and $y$ are column vectors, or is that yet to be defined? But to my main question: find $$\quad \frac{\...
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1answer
22 views

Level curves of exponential

I have the following two-variable function: f(x,y) = exp(-x^2-(y-1)^2) And I need to compute/sketch the level curves for ...
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1answer
35 views

When a mathematician says $f(y,x)$ is strictly increasing in $x$, what do they mean?

when someone says $f(x,y)$ is increasing in $x$, do they mean partial differentiation $\frac{\partial f(x,y)}{\partial x}$ or total differentiation $\frac{d f(x,y)}{d x}$? If $y=h(x)$, then $f(x,y) = ...
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Another difficult 2D trigonometric integral

This is a follow-up question to A difficult 2d trigonometric integral. Unfortunately, I had a mistake in my calculations and I need a different (yet similar) seemingly simple integral solved: $$\...
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1answer
53 views

differentiate $g(f(x),x)$ with respect to $f(x)$

Suppose I have a function $g(y,x)$ which is differentiable with respect to both arguments. I know that $y=f(x)$ is bijective, and thus the inverse function exists $x = f^{-1}(y)$. My question is when ...
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2answers
31 views

Prove that if $f(x, y)$ is continuous on $(0,0)$ then the function $g(x,y)=xyf (x,y)$ is differentiable on $(0,0)$

I made the following: $$\lim_{(x,y)\rightarrow(0,0)} \frac{xy f(x,y) - 0 -g_{x}(0,0) - g_{y}(0,0) (y-0)}{((x-0)^2 + (y-0)^2)^{\frac{1}{2}}}$$ $$=\lim_{(x,y)\rightarrow(0,0)} \frac{xy f(x,y)}{(x^2+y^2)^...
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1answer
48 views

A difficult 2d trigonometric integral

I'm trying to solve the following seemingly simple integral - so far, without success: $$\int_{a}^{b}\int_{a}^{y}\frac{\cos(x-y)}{xy}\mathop{\mathrm{d}x}\mathop{\mathrm{d}y}$$ For some $0<a<b$....
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2answers
82 views

Why do we have to try all the paths in multivariable limits?

I don't get the idea of trying "all" paths to know the value of the limit. The way I see it is I have to "scan the area" around the point, and that would be enought. for example, if all the straight ...
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2answers
24 views

Questions on changing bounds of integration for double-integrals

I'm having difficulty understanding how to properly change these bounds of integration. When I set up the bounds, which x or y bound is supposed to depend on the other? $$\int_{0}^{\pi/2}\int_{0}^{\...
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1answer
30 views

Proving limits existance

Im supposed to show and justify if the limit exists of this fucntion. $$\lim_{(x,y)\to(0,0)}\frac{\cos x-1-x^2/2}{x^4+y^4}$$ One way the solution says is fine is to approach along the line $y=0$ which ...
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2answers
26 views

Limit of double integral at origin

I have a problem set with the following problem: “Prove that, if $D_1={(x,y) | x^2+y^2 \leq r^2}$, then $\ lim_{r\to 0} \frac{1}{\pi r^2} \iint_{D_1} f(x,y) dA =f(0,0)$. I am just stuck, as I ...
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0answers
44 views

Is $f(x,y, z)$ considered three or four dimensional?

I am doing path integrals video for both scalar fields and there is some confusion on my part how dimensions are treated. $F(x,y) = z$ is considered $3$ dimensional in the video lecture as I see it....
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2answers
29 views

Proof convergence of vectors

I want to prove convergence of sequence. Let $a_k$ be a sequence in $\mathbb{R}^k$ with $a_k$=$(a_{ki}...a_{kn})$ for all k element of natural numbers, and let $a=(a_1,...a_n)$ element of $\mathbb{R^...
0
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1answer
27 views

2nd derivative of f(x(t) , y(t)) [duplicate]

So the first derivative w.r.t. t: $\frac{dz}{dt} = \frac{\partial f}{\partial x} \times \frac{dx}{dt} + \frac{\partial f}{\partial f} \times \frac{dy}{dt}$ How would I find the 2nd derivative? ...