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).

learn more… | top users | synonyms (1)

2
votes
1answer
13 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
votes
1answer
20 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?
2
votes
2answers
19 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}^ \! ...
0
votes
1answer
22 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
vote
1answer
14 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: ...
0
votes
1answer
18 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 ...
2
votes
1answer
26 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
votes
1answer
42 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.
2
votes
1answer
50 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$?
0
votes
2answers
18 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 ...
0
votes
1answer
12 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 ...
1
vote
2answers
30 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
votes
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) ...
0
votes
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 ...
0
votes
0answers
39 views

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
votes
1answer
42 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
vote
1answer
31 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
votes
1answer
23 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.
3
votes
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 ...
12
votes
0answers
111 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 ...
1
vote
2answers
25 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.
2
votes
2answers
19 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 ...
0
votes
1answer
22 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 ...
2
votes
4answers
174 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}) ...
2
votes
0answers
44 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. ...
5
votes
0answers
41 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 ...
1
vote
1answer
15 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 ...
1
vote
1answer
44 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
vote
1answer
15 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 ...
1
vote
2answers
27 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 ...
-2
votes
1answer
37 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 ...
1
vote
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.
0
votes
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
votes
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
54 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
vote
2answers
35 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 ...
1
vote
4answers
41 views

Double integral with variable change, why the $2\pi$?

I've seen a lot of examples from my textbook where the result of an integration is $2\pi$ instead of $0$, as I would expect it to be. And several of my results will match the correct result if I ...
2
votes
1answer
18 views

Hadamard's Lemma in multidimensional real analysis

This is Hadamard's Lemma: Let $U \subset \Bbb R^n$ be an open set, let $a \in U$ and $f: U \to \Bbb R^p$. Then the following assertions are equivalent. The mapping $f$ is differentiable at $a$. ...
0
votes
3answers
24 views

How do you know that rows are independent and what are the 120 terms?

I am having trouble with the question below, help me out;
0
votes
1answer
32 views

Line Integral where C is a line segment

The question states to evaluate the integral $\int_C \ xe^{y} ds$ from $(-1,2)$ to $(1,1)$ After finding the parametric equations $x(t) = 2t+1$ and $y(t) = -t+2$, solving for $ds$, and substituting, ...
0
votes
2answers
21 views

Line Integral: Using a line segment path

Evaluate the line integral $\int_C \sin(x)\sin(y)dx - \cos(x)\cos(y)dy$ where $C$ is the line segment fom $(0,-\pi)$ to $(\frac{3\pi}{2},\frac{\pi}{2})$ I parameterized and substituted into the ...
2
votes
2answers
32 views

Line Integral : Work done moving along a certain path

Find the work done by the force field $F(x,y) = -xi + 6yj$ along the path $C:y = x^3$ from $(0,0)$ to $(6,216)$ I tried parameterizing C which gave me $x(t) = t$ and $y(t) = t^3$ but do I use those ...
0
votes
2answers
52 views

Suppose $(x,y,z)$, $(1,1,0)$, and $(1,2,1)$ lie on a plane through the origin.

What determinant is zero? What equation does this give for the plane? I need some help here, am pretty stuck
0
votes
2answers
38 views

Double integral help?

How do I show that for $S = [0,1] \times [0,1]$ and $f(x,y) = \frac{x^2 - y^2}{(x^2+y^2)^2}$ that $\int \int_S f(x,y) dx dy \neq \int \int_S f(x,y) dy dx$ and why does this not contradict Fubini's ...
0
votes
3answers
51 views

Maxwell's Equations Divergence Question

$$ \left\{ \begin{align} \text{div } \textbf{E} & =0, \\ \text{div } \textbf{H} & =0, \\ \text{curl } \textbf{E} & = \frac{-1}{c} \frac{\partial \textbf{H}}{\partial t}, \\ \text{curl } ...
0
votes
1answer
37 views

Maxwell's Equations Curl Question

$$\left\{ \begin{align} \text{div } \textbf{E} &=0, \\ \text{div } \textbf{H} & =0, \\ \text{curl } \textbf{E} & = \dfrac{-1}{c} \dfrac{\partial\textbf{H}}{\partial t}, \\\text{curl } ...
0
votes
1answer
35 views

(Dis)continuity of function in $R^2$

$$f(x,y) = \begin{cases} a+2x^{2}-b(y-c), & x^{2}>2+x\wedge y<6\\ 3+cx-y, & else \end{cases}$$ $f(x,y)$ is continuous on $R^2$ if $a=-3, b=1, c=2$ I think it's true: insert ...
0
votes
0answers
13 views

Expression for volume without changing variables

My question is the same as this: Solving problem 3-29 in Spivak´s Calculus on Manifolds without using change of variables However, my solution, while makes perfect sense to me, is slightly ...
0
votes
1answer
30 views

Motivation for the definition of a conservative vector field

We say a vector field $\mathbf{F}$ is conservative if there exists some $f$ such that $\nabla f=\mathbf{F}$. I know that this implies that the field is path independent, and this makes sense to me: ...
1
vote
2answers
67 views

Triple Integral exercise

Calculate $\int\int\int_Dz\;dxdydz$ if $D$ is the region inside $z=0,z=\sqrt{x^2+y^2}$ and $x^2+y^2=1$. I would like to know if the answer I got is right. This is what I did: $(1)$ Change to ...