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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).

0
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0answers
6 views

Measure of $C^1$ path in $\mathbb{R}^2$

I started studying multivariable integration and still trying to grasp the conecpt of the measure. I`m doing excersices and I keep getting the feeling im doing something wrong so I hope one of you ...
0
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4answers
28 views

Absolute maximum for $f(x,y) = 4x^2 + 5y$

Suppose you have a region $R$ that satisfies $4x^2+y^2\leq 4$ What is the absolute maximum of the function $f(x,y) = 4x^2 + 5y$ on $R$? The correct answer is $10$ First, I calculated the critical ...
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1answer
11 views

Line Integrals of Vector Fields, Homework Conundrum

I am a student and I have a conflict with a given answer in the textbook. The question is the following: Evaluate the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ for the given vector field $\...
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2answers
19 views

$\frac{\partial{z}}{\partial{y}}$ if $z$ is given implicitly

Suppose $z$ is given implicitly as: $$e^z-x^2y-y^2z = 0$$ Find $$\frac{\partial{z}}{\partial{y}}$$. I let $F(x,y) = e^z-x^2y-y^2z$. Then, $$\frac{\partial{F}}{\partial{y}} = -x^2-2yz$$ $$\frac{\...
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1answer
24 views

Usage of Lagrange Multipliers in multivariable calculus

I just learnt about Lagrange Multipliers & am confused about why they are useful. Why can we not just check for critical points by checking if the gradient vector of the objective function $f$ is $...
1
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1answer
10 views

Compute mean and covariance matrix of $\bar{X}$ from a simple random sample

Given $\{X_\alpha , \alpha =1,...N\}$ a simple random sample obtained from any p-dimensional distribution with mean $\mu$ and covariance matrix $\Sigma$, compute the mean and the covariance matrix of $...
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0answers
5 views

Deduce normal marginal distribution from joint distribution

Given two random variables $X$ and $Y$ with joint distribution $F(x,y)=\phi (x) \phi (y) [1+ \alpha (1-\phi (x) )(1- \phi (y))]$ where $| \alpha | \leq 1$ and $\phi$ represents the standard normal ...
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0answers
14 views

Divergence theorem for vector field over top half of sphere [on hold]

$$F= 2xz^2i + (2(y^3)/3 + \tan z)j + (2(x^2)z + 5y^2)k.$$ Evaluate $∫FdS$ over the surface $S$, where $S$ is the upper half of the sphere centered at the origin, with radius $1$, oriented upwards.
3
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0answers
32 views

Looking for another way to calculate the integral $\iint_{D}{\sin(x)e^{\sin(x)\sin(y)}}\text{d}A$

Here, I have a little unpleasant way to calculate the following double integral $$\iint_{D}{\sin(x)e^{\sin(x)\sin(y)}}\text{d}A$$ where $D$ is the square area $D=\{(x,y)\in\mathbb{R}^2: 0 \le x \le \...
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0answers
17 views

How to prove that it is impossible to express one variable as function of others in a equation (implicit function theorem)?

Consider implicit function theorem. Since the conditions the theorem gives are only sufficient and not necessary I would like to know how can one prove that it is impossible to express (locally) one ...
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0answers
31 views

Intuition on Stokes' Theorem

Stokes' Theorem says that $$\int_C\vec{F} \cdot d\vec{r} = \int \int_S (curl \space\vec{F}) \space\cdot \vec{n} \space dS$$ I understand that $curl \space\vec{F}$ is the "spin" or "circulation" on a ...
1
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1answer
35 views

Show that there exists $T>0$ such that $\frac{e^{t}}{t^{1-\alpha}}-\int_{0}^{t}\frac{e^{\xi}}{\xi^{1-\alpha}}d\xi<0$ for all $t\geq T$

Let $\alpha\in\,]0,1[$. I want to show that there exists $T>0$ such that the positive function $F(t):=e^{-t}\int_{0}^{t}\frac{e^{\xi}}{\xi^{1-\alpha}}d\xi$ decays monotonically to zero. I ...
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1answer
30 views

The Jacobian of $(x,y)\mapsto (x+y^2,y+x^2)$ under the substitution $u=x+y^2$ and $v=y+x^2$.

I am given the map $(x,y)\mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated. (I am trying to find whether the ...
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1answer
30 views

Surface Area of Multiple Integration

Find the area of the upper half of the cone $x^2+y^2=z^2$ above the interior of one loop of $r=cos(2\theta)$. I know the formula for surface area is $\int_{x_0}^{x_1}\int_{y_0}^{y_1}\sqrt{(f_x)^2+(...
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0answers
27 views

Is $f(x,y) = \frac{x^ny^m}{x^{n'}+y^{m'}}$ continuous at origin? [duplicate]

Let function $\,\, f:\mathbb{R}^2\rightarrow\mathbb{R}\,$ be defined as $$ f(x,y) = \begin{cases} \frac{x^ny^m}{x^{n'}+y^{m'}}\text{,} &\quad(x,y)\neq(0,0)\\ \text{0,} &\...
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4answers
61 views

Does $\lim_{(x,y)\to(0,0)}\frac{6xy^2}{x^2+y^2}$ exist? [on hold]

I am trying to solve the following $$\lim_{(x,y)\to(0,0)}\frac{6xy^2}{x^2+y^2}$$ This is basically $0/0$ form, but as I saw other post that L'Hôpital's rule does not work on multiple variables. ...
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0answers
12 views

Intervals of a Multivariable Function

If the gradient at some point of a multivariable function equals $\vec{0}$, and the Hessian is positive or negative semidefinite, is there a notion, as in single variable calculus, of resolving the ...
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0answers
33 views

Find $du$ and $d^{2}u$

Find $du$ and $d^2u$, if $u = f(xyz, xy^2, xz^2).$ For $du$, I did $$du = f'_xdx + f'_ydy + f'_zdz$$ $$du =(f'_1yz+f'_2y^2+f'_3z^2)dx + (f'_1xz+f'_2*2xy)dy + (f'_1xy + f'_3*2xz)dz$$ How to proceed ...
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0answers
26 views

Derivative Of Infimum - Embedded Submanifold Of $\mathbb{R}^n$

This is a follow-up question to: Derivative Of A Function Defined In Terms Of An Infimum Let $A$ be a compact, (smooth) embedded submanifold of $\mathbb{R}^n$ and let $\Phi : \mathbb{R}^n \...
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1answer
31 views

Determine if a function is continuous in points

Determine if a function $f(x, y)$ is continuous in points (1, 1) and (0, 0). $$f(x, y) = \begin{cases} \frac{x^2+2xy-3y^2}{x^3-y^3}, & x \neq y \\ A, &x=y=0 \\ B, & x =y= 1\end{cases}$$ I ...
2
votes
1answer
22 views

Changing variables or coordinates?

While posing another question I got stuck on the distinction of the following two concepts; The components of a vector is usually referred to as it's coordinates, while trying to understand what "...
0
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1answer
26 views

How to study the critical points of a $2$-variable function?

I am revising some past exam questions and there is one that states: Study the critical points of the function: $$f(x,y)=x^2+y^2-x^4-y^4-2x^2y^2.$$ According to my professor, this is what I have ...
2
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0answers
34 views

Possible typing error in the book

One of the side effects of learning on your own is to seriously doubt if the book has an error or your reasoning is incorrect. My book on multivariable calculus (Rogawski 3rd Edition of Calculus ...
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2answers
27 views

Limits for triple integral parabolic cylinder

Determine the volume bounded by the parabolic cylinder $z=x^2$ and the planes $y=0$ and $y+z=4$. My work. I am not sure if I have the correct limits for this question. I used $x = -\sqrt{z},\dots, \...
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0answers
7 views

No compact surfaces with constant causal character in $\mathbb{L}^3$

The title is all there is to it: I want to know why there are no compact surfaces with constant causal character in $\mathbb{L}^3$. If possible, some indications in the computing of the causal ...
2
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0answers
128 views

Total variation on a creased surface

Suppose $S$ is a smooth surface without boundary and $f:S\to\mathbb R$ is continuous. Then, we can define the total variation of $f$ as: $$ \mathrm{TV}[f]:=\sup_{\|\phi\|_\infty\leq1} \int_S f(x)\...
3
votes
2answers
44 views

Triple integral of portion of cone (cylindrical polar coordinates)?

$V$ is the portion of the cone $z=\sqrt{x^2+y^2}\;$ for $\; x\geq 0.$ Find $$\iiint\limits_{V} xe^{-z} dV.$$ I am trying to solve this question. The answer is supposed to be just $4.$ I have worked ...
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0answers
10 views

Find the Fluid force on the vertical plate submerged?

Find the Fluid Force on a vertical plate submerged B. Categorizing everything i see, 1.its "submerged in sea water" below the surface 2.the y-axis Depth = h(y) = -y = -6 ft 3.the x-axis at the ...
1
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2answers
41 views

Find $\frac{\partial y}{\partial z}$ of the surface $g(s,t)=(s^2+2t,s+t,e^{st})$ near $g(1, 1) = (3, 2, e)$.

Consider the surface given by $g(s, t) = (s^2 + 2t, s + t, e^{st})$. Think of $y$ as a function of $x$ and $z$. Find $\dfrac{\partial y}{\partial z}(3,e)$ near $g(1, 1) = (3, 2, e)$. really ...
5
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1answer
89 views

find closed form for following double integral containing radicals

question: To prove : $\displaystyle\int_{0}^{1}\displaystyle \int_{0}^{1}\dfrac{dxdy}{\sqrt{1-x^2}{\sqrt{1-y^2}}{\sqrt{4x^2+y^2-x^2y^2}}}=\dfrac{3\left(\Gamma{\dfrac{1}{3}}\right)^6}{2^{\dfrac{17}{3}...
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0answers
33 views

How to evaluate surface integral?

I got 4/3pi as my answer. But it is probably wrong because it only accepts an integer, fraction, or an exact decimal. I set up my integral as 2 on the very outside of my two integrals. I have (0 to ...
3
votes
4answers
83 views

Does this limit exist on $\mathbb R^2$

$(x,y) \in \mathbb R^2$ $$\lim_{(x,y)\to(1,1)} \frac{(x-y)^{(x-y)}} {(x-y)}$$ Does the limit above exist? Neither I could compute it nor I could find directions which have different limit values. ...
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1answer
27 views

How can I calculate Fourier transform of this 3D function? [on hold]

Calculate Fourier transform of function $f:\mathbb{R}^3\rightarrow \mathbb{R}$ defined by this formula $$f(\mathbf{x})=\frac{1}{1+|\mathbf{x}|^2}.$$
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0answers
21 views

Integral of function undefined at one point

Let us consider a plane with polar coordinates. Let us also consider the following integral over any area $A$ on the plane: $$\iint_A f(r,\theta)\ \hat{r}\ dr\ d\theta\ $$ Here the function is $\...
4
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2answers
42 views

When taking double and triple integrals over some region, how do I know what order to place the integral bounds?

Is there a general way of approaching this or does this vary on a case by case basis?
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0answers
18 views

Why can't I use gradient vector's magnitude as surface integral's area element?

I understand that when calculating the surface integral, we have to parametrize the surface by a 2-variables vector equation. Then, the cross product of the two velocity partial vectors would then ...
0
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2answers
32 views

double integration of floor function

I'm stuck on a double integration problem: $$\iint ⌊x⌋*⌊y⌋ \,dA $$ over the region bounded by $x = -2$, $x = 1$, $y = 0$, $y = 2$. I know that the region formed is a rectangle and the integral ...
0
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1answer
19 views

Differentiating an inner product w.r.t matrices

Let $M_n(\mathbb{R})$ denotes the space of all $n \times n$ matrices identified with the Euclidean space $\mathbb{R^{n^2}}$. Fix a column vector $x \neq 0$. Define $f:M_n(\mathbb{R}) \rightarrow\...
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0answers
10 views

Finding a bound for a sum over multi indices

I want to show that $\sum_{|\alpha||,|\beta|\leq N}sup\{t^{\alpha}D^{\beta}(x_1-t_1)_{+}^{N+1}\cdot\cdot\cdot(x_n-t_n)^{N+1}_{+}|t_i>0 \text{ for all i=1,...,n}\} $ is bounded by something in the ...
1
vote
1answer
33 views

Finding if a path is tangent to a level surface

The question is as followed: Consider the path $r(t)=<t,2\sin2t,2\cos2t>$ TRUE/FALSE: the path $r(t)$ is tangent to the level surface $f(x,y,z)=x^2-y^2+z=1$ I guess what's throwing me off is ...
1
vote
1answer
27 views

Can I prove that a 2-variable limit does not exists if the limit on a curve is infinity?

Consider a 2 variable function $f(x,y)$ and the limit $$\lim_{(x,y)\to (0,0)} f(x,y)$$ If I find two continuous functions $\gamma_1(t)$ and $\gamma_2(t)$ such that $\gamma_1(0)=\gamma_2(0)=(0,0)$ ...
0
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1answer
25 views

Hydro Turbine Optimization

So I was doing the project related to Lagrange multiplier called hydroxyzine optimization recently and I have encountered problem on question4-6, which required me to plot a graph and find the optimum ...
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1answer
30 views

How to differentiate $\int_{B(t)} f(x,t) dx$ with respect to $t$?

$\int_{B(t)} f(x,t) dx$ is given and assume $x$ is an $n$ dimensional vector variable, $t$ is positive real variable and $f(x,t)$ is a sufficiently smooth function of both $x$ and $t$. Also $B(t)$ is ...
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votes
0answers
29 views

Differential of a function with absolute values

I am trying to calculate the differential of the following function around $(0,0)$: $$ f(x,y) = \begin{cases} x, & |x| > |y| \\ y, & |x| \le |y| \end{cases} $$ I know that if $f$ is ...
0
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0answers
15 views

Calculating the differential and relating it to the directional derivative

I need to calculate the differential of the following function, but I am struggling to make sense of the relationship between the directional derivative and differential: $f(x,y) = ln(x^2+2y+1) + \...
1
vote
1answer
60 views

Volume between cone and sphere of radius $\sqrt2$ with surface integral

Consider the cone $z^2=x^2+y^2$ between $z=0$ and $z=1$. Find the volume of the region above this cone and inside the sphere of radius $\sqrt2$ centered at the origin that encloses the cone. The ...
-1
votes
1answer
24 views

How to change from a double to a single integral changing variables?

So I have the following integral: $$I_1 = \iint u(x,y)dR$$ where $u(x,y)= e^{-(x^2 + y^2)}$ and the region $R$ is the rectangle $[-M,M]\times[-M,M]$. I need to prove that $I_1$ equals: $$I_2 = \...
0
votes
1answer
18 views

How do we conclude that $\nabla f(p) = 2np-\sum_{i=1}^n2p_i$?

I'm trying to show that $\bar{p}$ is a minimum Writing the function as $$f(p) =\sum_{i=1}^n\langle p-p_i,p-p_i\rangle,$$we have $$Df(p)(v) =\sum_{i=1}^n 2\langle p-p_i,v\rangle.$$So, write this ...
1
vote
1answer
22 views

Partial derivative of a two variables function, one of which dependent on the other

I found this exercise on the book of multivariable calculus from which I'm studying: "Find the partial derivative $\frac{\partial{z}}{\partial{x}}$ and the total derivative $\frac{\text{d}z}{\text{d}...
0
votes
0answers
18 views

Find the tangent space of Ellipsoid $M = \{(x,y,z)|\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\}$

Find the tangent space of $$M = \{(x,y,z)|\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\}$$ So I know the formula of tangent space for a manifold represnted by $F$ such that $F=0$: it is $ker (...