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
28 views

Surface/Path Integral Approach - Brain Fart?

Many times I have dealt with path and surface integrals of the following form $$\int_C \mathbf{F}\cdot d\mathbf{r} \,\,\,\,\,\textrm{(path integral)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int_S ...
2
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1answer
15 views

Showing the Existence of Total Derivatives

I was presented with the following problem regarding a function that has discontinuous partial derivatives: $$ f(x,y) =\begin{array}{lr} x y \sin(\frac{1}{x^2 + y^2}) : (x,y) \neq 0\\ 0 : (x,y) = ...
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1answer
21 views

Spherical coordinates from a given set

Describe the set in spherical coordinates. $$x^2+y^2=3z^2$$
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3answers
24 views

Parametrization of the intersection of two given surfaces

Find a parametrization of the intersection between the two curves $z=x^2-y^2$ and $z=x^2+xy-1$. I figure I should set them equal to each other but I'm not sure where to go from there: $$x^2-y^2 = ...
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0answers
3 views

If $f ' (x ; y)=0$ for every $x$ in open convex set, then $f$ is constant on open convex set.

$f′(x;y)=0$ for every $x$ in an open convex set $S$ and every $y$ in $\mathbb{R}^n$, Prove that $f$ is constant on $S$. $f′(x;y)$ is the derivative at $x$ in the direction $y$. Seems like I have ...
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0answers
11 views

Lie Derivative of a section on a vector bundle

I'm still trying to figure out how to do the Lie derivative of a Jacobian. (c.f. earlier unanswered post). If I know how how to do Lie derivatives on section of vector bundles, that would be ...
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0answers
13 views

construct with a wire of 18 cm a square and a circle, to reach the maximal amount of surface [on hold]

With a wire of length 18 cm construct a circle and a square. How much wire should be used for the circle if the total area enclosed by the figure(s) is Maximal?
3
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1answer
23 views

Proving that the iterated limit and the two dimensional limit are same

If $\lim_{(x,y)\rightarrow (a,b)} f(x,y) = L$ and if the one dimensional limits : $\lim_{x \rightarrow a}f(x,y)$ and $\lim_{y \rightarrow b}f(x,y)$ both exist, prove that : $$\lim_{x \rightarrow a} ...
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0answers
7 views

Show that $\frac{g^{N}(t)}{N!}=\sum_{|\alpha|=N}\frac{D^{\alpha}f(\vec{a}+t\vec{h})}{\alpha !}h^{\alpha}$

Let $f(\vec{x})$ be a $C^{\infty}$ function of $n$-variables. Let $g(t)=f(\vec{a}+t\vec{h})$. Show that $\frac{g^{N}(t)}{N!}=\sum_{|\alpha|=N}\frac{D^{\alpha}f(\vec{a}+t\vec{h})}{\alpha ...
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0answers
7 views

Differentiability of an extension of the 2-times continuously differentiable function on a half ball

Let $\Omega^+:=\{x=(x_1,\cdots,x_n)\in{\Bbb R}^n\mid |x|<1,x_n>0\}$ and suppose $f\in C^2(\overline{\Omega^+})$ with $f\mid_{\Omega_0}=0$ where $$ \Omega_0:=\partial\Omega^+\cap\{x_n=0\}. ...
3
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1answer
18 views

Differentiating a multivariable function

Knowing that $$z(x,y)=f(\frac{x}{y})$$I'm supposed to find $$x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y}$$ . This problem makes no sense to me, can anyone help with the ...
2
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2answers
66 views

Limit of a function with two variables: when do we stop looking for another value?

So for instance we have this limit of a function $\displaystyle\lim_{(x,y)\to(0,0)}{xy\over \sqrt{2x^2+y^2}}$, and the function isn't continuous at the point $(0,0)$. Now we can try to find the limit, ...
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1answer
30 views

Understanding directional derivatives

I am confusing myself when it comes to directional derivatives and gradients. The gradient of a function shows the direction of the greatest change. So when we chose a unit vector as the direction to ...
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0answers
12 views

Which of the following vector fields can be a gradient of a function?

Which of the following vector fields can be a gradient of a function? $A. F(x, y) = (2x + y, x + 2y)$ $B. F(x, y) = (2xy + y, x + 2y^2)$ $C. F(x, y) = (2x + y, x + 2)$ D$. F(x, y) = ...
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0answers
9 views

Is a mapping $\mathbb{R}^2 \to \mathbb{R}^2$, with harmonic components and positive Jacobian, injective?

Let $F = \left( \begin{array}{c} f \\ g \end{array} \right) \colon \mathbb{R}^2 \to \mathbb{R}^2$ be a mapping, each of its two components are harmonic functions in the plane, i.e. $\Delta(f) \equiv ...
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2answers
35 views

How can I parametize a curve of intersection of two surfaces?

To find out directional derivative $f(x.y.z)=x^2+y^2−z^2$ at $(3,4,5)$ along the curve of intersection of the two surfaces $2x^2+2y^2−z^2=25$ and $x^2+y^2=z^2$ I am trying to parametrize above two ...
0
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0answers
18 views

Prove $(\vec{r}\cdot\nabla)\vec{u}+\vec{r}\times (\nabla \times \vec{u})=\vec{r}(\nabla\cdot\vec{u})+(\vec{r}\times \nabla )\times\vec{u}$

Show that $$(\vec{r}\cdot\nabla)\vec{u}+\vec{r}\times (\nabla \times \vec{u})=\vec{r}(\nabla\cdot\vec{u})+(\vec{r}\times \nabla )\times\vec{u}$$ I have been trying to show this for the past few ...
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1answer
13 views

Cartesian equations for the line tangent to two surface.

I am asked to find a Cartesian equation for the line tangent to both the surfaces x^2+y^2+2z^2=4 and z=e^(x-y) at the point (1.1.1) I tried to find out normal vector to both surfaces and tangent ...
3
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2answers
39 views

Maximum / Minimum Question with 3 Variables?

I seem to be stuck in this problem, would need your help! Question: Assume I have : 147 of x, 174 of y, 238 of z A different amount of x, y ...
1
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0answers
17 views

Change of variables for functions with constraints

I want to find critical points of say F = (x1-x2) + (y1^2-y2^3) + (z1^2-z2^3) with constraints x1^1 + y1^2 + z1^2 -1 = 0 and x2^1 + y2^2 + z2^2 -1 = 0 I can do this by finding critical points of ...
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0answers
10 views

How would you find out the cartesian equation tangent to two surfaces at given point?

I am given two surfaces and asked to find out a pair cartesian equations that are tangent to these two surfaces. I know how to find out tangent plane to one surface. But, how would you find out a ...
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1answer
24 views

Evaluate the directional derivative along the curve of intersection of the two spheres..

I am given $f(x.y.z)=x^2+y^2-z^2$ at $(3.4.5)$ along the curve of intersection of the two surfaces $2x^2+2y^2-z^2=25$ and $x^2+y^2=z^2$ And evaluate the directional derivative. I know how to find ...
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0answers
46 views

A book like Michael Spivaks Calculus, for multivariate Calculus.

Is there a book like Michael Spivaks Calculus, that is for Multivariate Calculus? That is a "real analysis" multivariate calculus book?
2
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2answers
29 views

Evaluate 2D integral (by change of variable)

The question asks to evaluate integral $$\iint_D \Big[3-\frac12( \frac{x^2}{a^2}+\frac{y^2}{b^2})\Big] \, dx \, dy \ $$ where D is the region $$\frac{x^2}{a^2}+\frac{y^2}{b^2} \le 4 $$ I believe ...
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0answers
25 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 ...
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0answers
10 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 ...
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0answers
11 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 ...
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0answers
19 views

Calculate the flux coming out of a surface

Let F(x,y,z)=(2xy(z-2),x^2(z-2),x^2y) be a vector field and $\Sigma $ the surface defined as the portion of cone x^2+y^2=(z-2)^2 ...
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0answers
30 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
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2answers
58 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 ...
1
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1answer
28 views

Triple integral over a region [on hold]

My problem is to evaluate the following: $$\iiint_D{6xy\space dV},$$ where D is the solid, bounded from above by the plane $z=1+x+y$, below the region in the $xy$-plane, and by the curves ...
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0answers
26 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 ...
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1answer
17 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 $y$. ...
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1answer
23 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 ...
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1answer
27 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 ...
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0answers
18 views

Let $f$ be a scalar field such that $f ' (a ;-y)$ exsits [on hold]

Let $f$ be a scalar field where derivative of $f$ at point a with respect to vector $-y$ exists, $f '(a;-y)$ exists. Is it always true for any nonzero vector $y , f '(a;-y) = - f '(a;y)$?
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1answer
12 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 ...
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1answer
25 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,
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1answer
19 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, $f ' (a;y)$ ...
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1answer
20 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 ...
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2answers
16 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.
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1answer
23 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 ...
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0answers
10 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 ...
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0answers
7 views

mean value theorem for scalar field

I just want to make sure that Mean value theorem for scalar field works same as one- dimensional mean value theorem. Usually, my book explains Mean value theorem for scalar field on interval [0.1]. ...
2
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0answers
6 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 and for every x in ...
2
votes
1answer
31 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)= ...
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0answers
12 views

Can anyone help me prove derivative of scalar field using mean value theorem?

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 and for every x ...
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0answers
22 views

There is no scalar field such that $f '(a)>0$ for fixed $a$ and for every nonnegative vector $y$ [on hold]

I am trying to prove this. But can't think of how I should start. Anyone has some ideas? and why is there a scalar field $f'(a)>0$ for every $a$ and for fixed vector $y$ ? can anyone give me an ...
1
vote
1answer
37 views

Find limit of $f(x,y) = \frac{x^3\cdot y-x \cdot y^3-x}{1-x \cdot y}$ where $(x,y)$ approaches $(0,0)$ [on hold]

$$ \lim\limits_{(x, y)\to (0, 0)}\frac{x^3y-xy^3-x}{1-xy} $$ I don't know how to solve this limit. All help is appreciated.
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0answers
20 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 ...