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
38 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
3
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1answer
33 views

Sketching a surface

I'm trying to draw a sketch to get a feel of the situation but am confused as to what the question is asking. I have sketched $y^2=8x$ in the plane $z=0$ and marked on the points where $y$ and $z$ ...
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1answer
30 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
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3answers
80 views

How do I find a point on the surface of a sphere

How do I find a point on a sphere knowing its radius and center point ? I have a sphere: $$x^2+(y-1)^2+(z+3)^2=16$$ Obviously its center point is $(0,1,-3)$ and its radius is $4$. I am asked to find ...
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2answers
27 views

Parametrization Question

When computing a line integral, or any integral that requires parametrization, what are you integrating with respect to? For example, if parametrizing in polar coordinates, with $x=r\cos\theta$ and ...
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1answer
16 views

Average value for multiple integrals

If there is a function $f(x,y)$ and we want to find the average value over a region $R$ defined by $0<x<1$ and $0<y<x$, how is that computed? I know that it would be something like this: ...
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0answers
17 views

Minimization with two functions that are not completely related

Two caveats: 1) This is a problem I formulated myself, and so may not be structured correctly/logically. 2) I don't have an extensive math background, but am currently finishing up Calc 3. I have an ...
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1answer
33 views

Use triple integrals to integrate over a tetrahedron

Integrate $f(x, y, z) = x^2 + y^2 - z$ over the tetrahedron with vertices $(0, 0, 0), (1, 1, 0), (0, 1, 0), (0, 0, 3)$. I need to use triple integrals to solve this, so I made a diagram and set the ...
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2answers
33 views

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial ...
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1answer
60 views

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

I came across the following when studying a proof related to the normal distribution: $$\left(\int_{-\infty}^{\infty}e^{-z^2/2}\ dz \right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 ...
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1answer
11 views

Reference for transformation of integrals over Lipschitz boundaries

Let $D\subseteq\mathbb{R}^d$, $d\ge 2$, be a bounded Lipschitz domain. Then according to page 314 of [Function spaces, Alois Kufner, 1977] one can define a surface integral of a real valued function ...
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2answers
74 views

Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$

I need to evaluate the following integral: $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(x^2-xy+y^2)}dx\, dy$$ I thought of evaluating the iterated integral ...
1
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1answer
22 views

Tangent line to a curve statement

I am having problems understanding some parts of the proof of some statement related to tangent line to a curve. I'll copy the exact statement and proof and then my doubts. Statement If $\mathcal C$ ...
3
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0answers
30 views

Does chain rule require continuously differentiability?

Recently I read the book Advanced Calculus written by Fitzpatrick. The Theorem 15.34 tells that If $F:\mathbb R^n\to\mathbb R^m$ is CONTINUOUSLY differentiable (all partial derivatives exist and ...
3
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2answers
35 views

how to prove gradients vectors are the same in polar and cartesian coordinates.

Suppose $T=T(r,\theta)=G(x,y)$ How do you prove $\nabla T(r,\theta)=\nabla G(x,y)$? I can think of some arguments in favor of this equality, but I want an actual proof or a very good intuitive ...
2
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1answer
22 views

Transformation rule for partial derivatives

I can't fathom the step I have highlighted in green. Am I using the chain rule in 3 dimensions? What is it that I am transforming here?
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2answers
23 views

Confusion about Spherical Coordinates Transformation

We have a function $$f(x,y,z) = \frac{e^{-x^2 -y^2 -z^2}}{\sqrt{x^2+y^2+z^2}}$$ and we want to integrate it over the whole $\mathbb{R}^3$. Then what i got is the following: $$\int_{\mathbb{R}^3}^ \! ...
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1answer
25 views

Directional Derivative Derivation

I don't understand the part underlined in the derivation of the directional derivative. Why is the $\lim_{Q \to P}$ interchangeable with $\lim_{N \to P}$? I understand that the surfaces are getting ...
1
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1answer
15 views

Evaluate $\int_C \nabla(r^4) \cdot \hat n ds$ in terms of moments of inertia…

Curve $C$ is closed plane curve, $a$ and $b$ are the moments of inertia about $x$ and $y$ axes, $\hat n$ is the unit outward vector and $r = \left|x \hat i + y \hat j\right|$. Here's what I have: ...
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1answer
20 views

Calculating the area of a region using a mapping

The region: $\{{(x,y) \mid x^{2} < y < 2x^{2}, 2y^{2}<x<3y^{2}, x > 0, y > 0}\}$ The mapping: $u = y/x^{2}$, $v = x/y^{2}$ I calculated the jacobian to be $\frac 34$ which means ...
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1answer
30 views

Parametrizing to Calculate Flux

Evaluate the flux of $\mathbf{f}$ across the oriented surface $\Sigma$ by computing the surface integral $\iint_{\Sigma} \mathbf{f} \cdot d\sigma$, where $\Sigma$ is the surface $z=xe^y$ for $0 \leq x ...
3
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1answer
45 views

Area of a Curved Surface

Find the area of the part o the surface $z=xy$ that lies within the cylinder $x^2+y^2=1$. I'm not sure how to set up the surface integral to compute this.
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0answers
53 views

Existence of a polynomial with a certain property [duplicate]

Does there exist a polynomial $P(x,y)$ s. t. $\forall x \in \mathbb{R} \,\,\forall y \in \mathbb{R} \,\,P(x,y)>0$ and $ \inf_{x \in \mathbb{R},\,y \in \mathbb{R}} P(x,y)=0$?
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2answers
19 views

First order approximation of multivariable function

Let $O$ be an open subset of $\mathbb R^2$ and suppose the function $f:O\to\mathbb R$ is continuous at the point $(x_0,y_0)$ in $O$. Define tangent plane as $\phi(x,y)=a+b(x-x_0)+c(y-y_0)$ where ...
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1answer
15 views

Triple integral visualisation problem with a sphere and a cylinder

Write a triple integral in cylindrical coordinates for the volume of the solid cut from a ball of radius 2 by a cylinder of radius 1, one of whose rulings is a diameter of the ball. I am unable to ...
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2answers
33 views

A curious question about optimizing a function of 2 variables.

Let $f(x,y)$ be defined and has continuous first and second partials on a domain $D$. Also, let $$A = \frac{\partial^2 f}{\partial x^2} \\ B = \frac{\partial^2{f}}{\partial x \partial y} \\ C = ...
2
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1answer
10 views

If $\vec r = x \hat i + y \hat j + z \hat k$ and $r =| \vec r |$ show that $curl [f(r) \vec r] = 0$

I know that $\nabla \times f(r) \vec r = \nabla f(r) \times \vec r + f(r) \left ( \nabla \times \vec r \right )$. I figured that the rightmost expression is $0$. How do I prove that $\nabla f(r) ...
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1answer
54 views

Help with double integral

I need to prove if this integral exist (and some others) but i would like to know if there is a condition to say if the integral exist (for example in this case) that would help me solve this kind of ...
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0answers
68 views
+50

quasi-convexity of a function

Can someone help me identify whether the following function is quasi-convex? Let $p>1$. For $x=(x_1,\dots,x_n),x_i>0,\sum_ix_i=1$, we define $$f(x) = -\log \sum_i (x_i/\|x\|_p)^{p-1}.$$ Plots ...
3
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1answer
45 views

Use a double integral in polar coordinates to find the area

So the area is just an intersection of two circles Converting the two circles to polar coordinates, I get: $r(r-2\sin\theta)=0$, and $r(r-2\cos\theta)=0$ Ummm so $r =0$ and r = $2\sin\theta$ ...
1
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1answer
34 views

How to factor and reduce a huge determinant to simpler form? Linear Algebra

So, I have learned about cofactor expansion. But the cofactor expansion I know doesn't reduce the number of rows and colums to one matrix. I usually pick a colum, multiply each element in the column ...
0
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1answer
24 views

Area of a Paraboloid inside a Cylinder

Find the area of the part of the paraboloid $x=y^2+z^2$ that is inside the cylinder $y^2+z^2=9$. I'm not sure how to set up the integral to compute this. Thanks.
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2answers
41 views

Area of spherical cap with integrals

Given a sphere $S$ of fixed diameter $D$ (or radius $R=D/2$, it will be convenient to have both, I suppose), and a point $P$ on its surface, let's create a ball $B$ of variable radius $r$ with its ...
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0answers
123 views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\left\{\frac{1}{x_{1}\cdots x_{n}}\right\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_{0}^{1} \! \cdots \! \int_{0}^{1} \left\{\frac{1}{x_{1}x_{2} \cdots ...
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2answers
26 views

Equation of a Tangent Plane

Find the equation of the tangent plane to the given surface at the given point. $x=u^2, y=v^2, z=uv$ at $u=1, v=1$ How would you find the tangent plane when the surface is in this format? Thanks.
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2answers
21 views

Question on Green's Theorem

Consider the vector field $\textbf{f}(x,y)=(ye^{xy}+y^2\sqrt{x})\textbf{i}+(xe^{xy}+\frac{4}{3}yx^{\frac{3}{2}})\textbf{j}$. Use Green's Theorem to evaluate $\int_C\textbf{f} \dot d\textbf{r}$, where ...
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1answer
24 views

Proving or disproving continuity of a function

Consider a function $f:\mathbb{R}^{n}\times \mathbb{R}^{+} \rightarrow \mathbb{R}$, with the property that for a fixed vector $a:=(a_1,a_2,\cdots,a_n) \in \mathbb{R}^{n}$, there exist a finite ...
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4answers
176 views

Line Integral Around a Triangle

Let $R$ be the interior of the triangle with vertices $(0,0), (4,2),$ and $(0,2)$. Let $C$ be the boundary of $R$, oriented counterclockwise. Now evaluate the integral below. $$\int_C(y+e^\sqrt{x}) ...
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0answers
45 views

Proving a set is of measure zero.

Let $C\subset A\times B$ be a set of content zero. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B: (x,y)\in C\}$ is not of content zero. Show that $A'$ is a set of measure zero. ...
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0answers
44 views

How to take limit *along a path*

So in multivariable calculus for a limit of a function to exist, the limits of the function along all possible paths must exist and equal the same value. But how does one calculate the limit along a ...
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1answer
16 views

x-partial of $f(x,y) = (xy)/(x^2 + y^2)$ exists at (0,0)?

Let $f(x,y)= \begin{cases} \frac{xy}{x^2 + y^2}, & (x,y) \ne (0,0) \\ 0, & (x,y) = (0,0) \\ \end{cases}$. a) Show that $\frac {\partial f}{\partial x}|_{(0,0)}$ and $\frac {\partial ...
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1answer
45 views

Rotated arc in $\mathbb{R}$ [on hold]

We have got the arc $$A = \{(x,y) \in \mathbb{R} \mid x^2 + y ^2 = R ^2, 0 \leq x \leq R, 0 \leq y \leq R\}$$ and $R$ is positive real number. What is the area of ​​rotational figure obtained if ...
1
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1answer
16 views

derivative after composition with linear map

Let $f: \mathbb{R}^3 \to \mathbb{R}$ be a polynomial function and let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be an invertible linear map. If $\nabla f(P) \neq 0$ for all $P \in \mathbb{R}^3 - \{0\}$ does ...
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2answers
33 views

derivatives of a vector of functions with respect to a vector

Let $\vec W \in \mathbb R^3$. What is the general solution to: $$\frac{\partial}{\partial \vec{W}} \begin{pmatrix} f(\vec W) \\ g(\vec W) \end{pmatrix} $$ I think that in the ...
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1answer
38 views

Find the average temperature for the following regions [on hold]

Consider the temperature function $T(x, y, z) = \large\frac{z}{1+x^2+y^2}$ where there is a heat source along the $z$ axis increasing in temperature as you get farther away from the origin. Find the ...
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3answers
23 views

Find the volume of the solid enclosed by the paraboloids

Find the volume of the solid enclosed by the paraboloids $z=16-3x^2-3y^2$ and $z=4$ so what i did is this $4=16-3x^2-3y^2$ and I'm not sure about the following steps.
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0answers
22 views

Checking if the Hessian is the derivative of the gradient

Suppose f: R^n --> R. I have a code that computes the gradient of f. I have another code that computes the Hessian of f times a vector. Now I want to check if they are correct. Specifically, I am ...
0
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1answer
12 views

third “power” differential in vector calculus

What is the meaning of the following in vector calculus: $$ d^3\textbf{r} $$ where $\textbf{r}\in R^3 $? For example, it is used sometimes in the definition of the electric dipole moment ...
2
votes
1answer
59 views

Differentiability at a point

Let $f:\mathbb{R}^{2}\mapsto\mathbb{R}\mathbb{}^{2}$ be given by $$f(x,y) = \left(\begin{array}{c} x^{2}y+2y-x\\ 3xy+4y \end{array}\right)$$ Find a open set containing (0,0) where f has a ...
1
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2answers
38 views

Is $\nabla \cdot (\mathbf{a} \mathbf{b}^\mathrm{T}) = \mathbf{b}(\nabla \cdot \mathbf{a})+(\mathbf{a}\cdot \nabla) \mathbf{b}$ correct?

Wikipedia says that the following statement is a vector identity: $$\nabla \cdot (\mathbf{a} \mathbf{b}^\mathrm{T}) = \mathbf{b}(\nabla \cdot \mathbf{a})+(\mathbf{a}\cdot \nabla) \mathbf{b}$$ Where ...