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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2
votes
0answers
75 views

Prove that exists $\epsilon >0$ such that $S\cap C\cap B((0,0,0),\epsilon)=\{(0,0,0)\}$

I can't find the way to do this exercise. We consider $S=\{(x,y,z) \in \mathbb{R^3}: f(x,y,z)=0 \}$, where $f$ is a $C^1$function on $\mathbb{R^3}$ such that $f(0,0,0)=0$ and $\nabla ...
2
votes
0answers
27 views

Parametrisation of surface

Let $K= \{ (x,y,z) \in \mathbb{R}^3 : \sqrt{x^2+y^2} \leq z,\,\, x^2 + y^2 + z^2 = 1 \}$. I need a parametrisation of $K$ in order to calculate the flux of some function through $K$. I'm not sure ...
1
vote
0answers
14 views

Is this the proper way to differentiate a vector with scalar?

$\vec{r}(t)=(r_0+kt)\cdot\begin{pmatrix}\sin(\omega t)\\\cos(\omega t)\end{pmatrix}$ $\vec{r}(t)=\begin{pmatrix}r_0\sin(\omega t)+kt\space \sin(\omega t)\\r_0\cos(\omega t)+kt\space \cos(\omega ...
2
votes
1answer
48 views

Determine the volume of $A:=\{(x,y,z)\in \mathbb R^3 : \sqrt{x^2+y^2}\leq f(z)\}$

Let $f\in L^2(\mathbb R)$ and $f\geq0$. Determine $A:=\{(x,y,z)\in \mathbb R^3 : \sqrt{x^2+y^2}\leq f(z)\}$. "Normal" substitution $(x=rcos(\phi),y=rsin(\phi))$ did not help a lot, since I dont have ...
2
votes
2answers
285 views

Area enclosed by cardioid using Green's theorem

Let $$\gamma(t) = \begin{pmatrix} (1+\cos t)\cos t \\ (1+ \cos t) \sin t \end{pmatrix}, \qquad t \in [0,2\pi].$$ Find the area enclosed by $\gamma$ using Green's theorem. So the area enclosed by ...
0
votes
0answers
62 views

Absolute square in deriving Fourier transform variance

I'm having some trouble understanding how to derive the variance of the Fourier transform. This is for an image, i.e., it's a 2D transform. The variance is $|\hat{I}(\xi,\eta)|^2$, the absolute ...
0
votes
1answer
20 views

If $∇f(a)\cdot y ≤ 0$ for every vector $y$, why does $\nabla f(a)$ have to be zero?

If $f$ is differentiable at every point in $B(a)$ and $f(x)≤f(a)$ for all $x$ in $B(a)$, prove that $∇f(a)=0$. I actually did some work and found out that $∇f(a)\cdot y ≤ 0$ for every vector $y$. ...
0
votes
1answer
28 views

How to determine a function whose minima falls on a specified curve?

I have a family of curves given by $g(x,y)=C_0 yx^{-n}$. How can I determine the function $f(x,y)$ for the family of curves that satisfies the condition that the local minima $\frac{\partial ...
0
votes
1answer
34 views

changing the order of double integration

I am a little bit stucked about the changing the order of a double integration. $\displaystyle \int_{0}^{\infty}\int_{t-n}^{t}f(n,s)dsdn$ I try to represent the upper and lower bounds by a graphic ...
0
votes
1answer
34 views

Image of a circumference by a vector function

I'm doing this exercise and I don't know how to finish. Consider the vector function $F(x,y) = (x^2+y^2, 2xy)$. Determine the image of the circumference $x^2+y^2 = a^2$, $a>0$, and obtain the ...
-1
votes
1answer
47 views

Assume $f$ is differentiable at every point of $B(a)$ and $f(x)$ is less than or equal to $f(a)$

Over the scalar field, If $f$ is differentiable at every point in $B(a)$ and $f(x)$ is less than or equal to $f(a)$, prove why gradient of $f(a)$ is $0$. Just don't understand how to start with,
3
votes
3answers
157 views

Why Lagrange multipliers don't help to find the minimum of $f(x,y)=x^2+y^2$ with the constraint $y=1$?

Please help me understand why the following doesn't work. Say I want to find the minimum of the function $f(x,y)=x^2+y^2$ with the constraint $y=1$. So I declare the helper function ...
0
votes
1answer
82 views

Derivative over scalar field with respect to fixed point proof.

Prove there is no such scalar field that $f '(a;y) >0$ for fixed point $a$ and every non-zero vector $y$. I posted this question but some of you pointed out that it is not clear. So, $f ' (a;y)$ ...
1
vote
1answer
28 views

Let $f: \mathbb R \times \mathbb R^n \rightarrow \mathbb R^n$ given as $f(t,y)$. How I understand:“$f$ has continuous partial $y$-derivatives”?

Suppose I have a function $f: \mathbb R \times \mathbb R^n \rightarrow \mathbb R^n$ given as $f(t,y)$. Then how should it be understood "$f$ has continuous partial $y$-derivatives" ? Should it be ...
0
votes
2answers
34 views

plotting of level curves using pen and paper

I want to plot the level curve for the function $f(x,y)=\frac{y^2-x^4}{y^2+x^4}$ . I tried by substituting $f(x,y)=k$. But I am Unable to draw it using paper and pen. Kindly help me.
1
vote
1answer
55 views

multivariable limit problem

I have a confusion regarding this problem. Problem: $\displaystyle f(x,y)=\frac{\sin^2|x+2y|}{x^2+y^2}$ is continuous for all $(x,y)\neq (0,0)$. True or false? I think that the limit does not exist ...
1
vote
0answers
35 views

Double integral discontinuity question

In lecture we were doing the problem $\int {1 \over \sqrt{xy}}\,dxdy$ over the region $[0,1] \times [0,1]$. Since the function is undefined when $x=0$ or $y=0$ we took the limit as $\delta,\epsilon ...
3
votes
1answer
203 views

Mean value theorem and scalar field proof

Assume that $f′(x;y)=0$ for every $x$ in some $n$-ball $B(a)$ and for every vector $y$. Use the mean value theorem to prove that $f$ is constant on $B(a)$. And if $f′(x;y)=0$ for a fixed vector $y$ ...
2
votes
1answer
56 views

Finding the unit normal vector

Q. Consider the following vector function. $$ r(t)= \langle 6\sqrt{2}t,e^{6t},e^{-6t} \rangle $$ Find the unit tangent and unit normal vectors T(t) and N(t). I found $$T(t)= ...
1
vote
1answer
177 views

Volume of $y = 6\sqrt{\sin(x)}$ rotated around $y$-axis using triple integrals

The problem is to find the volume of $y = 6\cdot \sqrt{\sin (x)}$ rotated around the $y$-axis when $0 \leq y \leq 6$. I know this can be done by the sv-calc method of volumes of revolution but I ...
3
votes
1answer
110 views

How can I find line segment connecting two vectors?

Let $S$ be a subset of $\mathbb{R}^n$. it is called convex if for all pairs of $a$, $b$, line segment from $b$ to $a$ is element of $S$. And it is given that $at+(1-t)$ is line segment between two ...
0
votes
3answers
56 views

Multivariable: Show the limit is $0$.

I already proved that for $\alpha =1$ the limit doesn't exists. Now I need to show that for $\alpha > 1$ the limit does exists and equals $0$. $$\lim_{(x,y)\to (0,0)} \frac{\left|x\right|^\alpha ...
0
votes
1answer
58 views

Solution of a Partial Differential Equation

Problem statement Solve $\frac{\partial f}{\partial x}-x\frac{\partial f}{\partial y}=y$ using the change of variables $\left\{\begin{matrix} u=ax^2+y \\ v=x \end{matrix}\right.$ for a suitable ...
2
votes
1answer
62 views

Find $F'(t)$, where F is an integral

I need to find $F'(t)$, where $F(t)=\int_{[0,t]^2}e^{\frac{tx}{y^2}}dxdy$. My first approach: Let's observe that $\int e^{\frac{tx}{y^2}}dx=\frac{y^2}{t}e^{\frac{tx}{y^2}}+C$. So I get: ...
1
vote
1answer
29 views

Cylindrical limits of integration for a particular triple integral

In cylindrical coordinates, what would be the limits of integration for the triple integral serving to find the volume of the region in $\mathbb R^3$ bounded by: $x^2 + y^2 = y$ and the sphere of ...
1
vote
2answers
201 views

Find all planes which are tangent to a surface

I'm given the surface $z=1-x^2-y^2$ and must find all planes tangent to the surface and contain the line passing through the points $(1, 0, 2)$ and $(0, 2, 2).$ I know how to calculate tangent planes ...
2
votes
2answers
167 views

Can every differentiable scalar function be written as a divergence of some vector field?

My question is simple: can every differentiable function $f$ defined on a bounded, connected subset of $\mathbb{R}^3$ be written as a divergence of some vector field ? That is, given the vector field ...
0
votes
1answer
38 views

Rectangles in one dimension

I have to prove the following proposition : Show that the intesection of two rectangles in $\mathbb{R}^{n}$ is either the vaccum or is another rectangle. My attempt: I one is embeded in the other ...
1
vote
1answer
39 views

Interpreting limit notations

My question is: Are the following notations equivalent or not: $$(1)\;\;\;\;\;\;\text{When}\;||\textbf{x}||\rightarrow 0,\;\text{then}\;\;\;\frac{f(\textbf{x})}{||\textbf{x}||}\rightarrow0$$ ...
1
vote
1answer
28 views

$F(x,y)=2x^4-3x^2y+y^2$. Show that $(0,0)$ is local minimum of the Reduction of F for every linear line that passes through $(0,0)$.

first of all I checked if (0,0) is critical point $Df(0,0)=(8x^3-6xy,-3x^3+2y)| = (0,0) $ now my idea was to replace $y$ with $xk$ because of the reduction of $F$ ,and find the hessian matrix to ...
1
vote
1answer
56 views

Proving that $2$-D parabolic coordinates are orthogonal

How can we prove that the parabolic coordinate system in two dimensions is orthogonal? I tried using the dot product, but don't know where to start or what basis vectors can be used in two dimensions. ...
0
votes
0answers
27 views

Can this multivariable function exist?

(3) Is there a function of two variables whose z = 0 level curve consists exactly of the circles $x^2$ + $y^2$ = 4 and $x^2$ + $y^2$ = 10? If so, what is an example? If not, why not? I initially ...
1
vote
1answer
184 views

Independepnce of path in a closed curve line integral

Let $f(t)$ be a continuous function. Let $C$ be a smooth closed curve. Show that $$\oint\limits_C xf(x^2 + y^2)\,dx + y f(x^2 + y^2)\,dy = 0$$ Hint: Remember that $f(t)$ has a primitive function ...
2
votes
1answer
48 views

Proof of transformation law for double integrals

The second volume of Apostol's Calculus seems rather circumspect in its discussion of the change of variables formula for double integrals. Section 11.29 offers a proof under the following very ...
5
votes
4answers
319 views

Chain rule for partial derivatives intuition

Can somebody give me an intuitive explanation for the below equations. I'm not sure how they come about and how they can be perceived logically. $$\frac{\partial z}{\partial s} =\frac{\partial ...
1
vote
0answers
41 views

Kneser Inequality in multivariables

Based on the Kneser Inequality ("Polynomials and Polynomial Inequalities", p. 260) one has $\Vert q \Vert_{[-1, 1]} \Vert r \Vert_{[-1, 1]} \leq C(n, m) \Vert q r \Vert_{[-1, 1]}$ where all norms ...
3
votes
2answers
1k views

Parametric equations for hypocycloid and epicycloid

Suppose that the small circle rolls inside the larger circle and that the point $P$ we follow lies on the circumference of the small circle. If the initial configuration is such that $P$ is at ...
1
vote
2answers
177 views

Choosing path to show limit does not exist

I'm trying to show that the limit as $(x,y)$ go to $(0,0)$ for the function $f(x,y) = sin( x + y )/( |x| + |y|)$ does not exist. I initially tried the path $y=2$ and $y=1$, but I don't think I can use ...
1
vote
1answer
56 views

Finding flux across surface

Let S be surface {$(x,y,z) : x^{2} + y^{2} + 2z =2 . z \geq 0 $} Given F = $(y,xz,x^{2}+y^{2})$ n is outward normal .I have to find net flux through S . Since its closed surface so i applied Gauss ...
0
votes
0answers
29 views

About the $d\mathbf{s}$ notation

Let $\mathbf{F}$ be a vector field that changes with time, that is, written in components:$$\mathbf{F}(\mathbf{x},t)=(F_1(\mathbf{x},t),F_2(\mathbf{x},t),F_3(\mathbf{x},t))$$ where ...
2
votes
0answers
49 views

How to compute the covariance matrix of a random variable uniformly distributed in an ellipsoid

Suppose that x is a random variable uniformly distributed in an ellipsoid \begin{equation} x^{T}Mx\leq\delta, \end{equation} where $x\in \mathbb{R}^{n}$. Clearly, the mean of $x$ is zero. The ...
2
votes
0answers
18 views

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. [duplicate]

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. Here $P(1,-1,7)$ and $Q(7,5,1)$. I have tried to find $r(t)$ by using the formula $r(t)=p+t(p-q)$ but ...
1
vote
2answers
144 views

When does a double integral represent a surface area, and when does it represent a volume?

When does $\int_Af(x,y)dA$ represent a surface area geometrically, and when does it represent a volume? In my lecture notes I'm told it represent the volume underneath the surface $z=f(x,y)$, but I've ...
2
votes
3answers
62 views

What does it mean for a function to increase along a curve?

I think that if we were to say that, for instance, $y$ increases along the curve, (with no specific rate) then this means for the derivative to simply be positive. Or does it mean to choose the ...
1
vote
0answers
54 views

line integrals explanation

I am very new to this so sorry if it is obvious. Compute the line integral $\int Fdr $ where $F(x,y)=(x^2y,y^2x)$; $r(t)=(\cos t,\sin t)$; $t\in[0,2\pi]$. So what I would do is find $r'(t)=(-\sin ...
1
vote
1answer
56 views

Is $f:\mathbb{R}^2\to\mathbb{R}$ differentiable on $(0,0)$?

Let $f:\mathbb{R}^2\to\mathbb{R}$ such that $f(x,y) = (x^2+y^2)\sin(\frac{1}{x^2+y^2})$ for $(x,y)\ne (0,0)$ and $f(x,y)=0$ for $(x,y)=(0,0)$. Is $f$ differentiable on $(0,0)$? So let's first ...
0
votes
1answer
27 views

Parametrize given curves

I'm given the following curves: $x = y^2 + 1$, $z = x + 5$ I'm eventually trying to find the unit tangent vector, so I need to find the r vector. Could I just assign $y = t$, and then have $<t^2 ...
1
vote
1answer
55 views

How to find the normal vector in a TNB problem

I have done this TNB problem multiple times; however, my online homework system keeps telling me my answer is incorrect. I was hoping someone would look at my work and tell me where I'm going wrong? ...
0
votes
1answer
20 views

continuity with 2-variables

The question is Determine whether $f$ can be defined at $(0,0)$ so that is is continuous $$f(x,y) = \frac{x^py^q + x^ry^s}{x^qy^p + x^sy^r}, p,q,r,s > 0$$. I chose numbers for p,q,r,s and ...
0
votes
1answer
36 views

Computing a specific line integral

Here is the problem as I have been given it: A curve $C$ is given in Cartesian coordinates by $r(t) = (cos(sin(nt))cost,\; cos(sin(nt))sint,\; sin(sin(nt)))$, with $t$ between $0$ and $2$$\pi$ ...