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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3 views

Reparametrization of a function

Given \begin{equation} S(t)= \left( 1- \left( 1-{{\rm e}^{- \left( \rho\,t \right) ^{k}\theta}} \right) ^{\gamma} \right) ^{\frac{1}{\theta}} \end{equation} where $\rho,k,\theta$ and $\gamma$ are ...
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1answer
11 views

Find the surface area of that part of the cylinder

Find the surface area of that part of the cylinder given by $\mathbf{r}(u,v) = 3\cos u \mathbf{i} + 3\sin u\mathbf{j} + v\mathbf{k}$ over the region where $0\leq u \leq2\pi$ and $0\leq v \leq2$. The ...
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1answer
59 views

Volume of the ellipsoid $(x+2y)^2+(x-2y+z)^2+3z^2=1$

Find the volume of the ellipsoid $(x+2y)^2+(x-2y+z)^2+3z^2=1$, using integration. It is clear that this is not centered at the origin. So, how do I find the limits for an integral? Any suggestion ...
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1answer
15 views

Parameterize $\{(x,y,z) \in \mathbb{R}^3 \colon (\sqrt{x^2 +y^2} -3 )^2+z^2 = 1\}$

I need to parameterize the surface $$S=\{(x,y,z) \in \mathbb{R}^3 \colon (\sqrt{x^2 +y^2} -3 )^2+z^2 = 1\}.$$ My hint is that $S$ is a torus. I barely know where to begin. I have some idea on perhaps ...
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1answer
46 views

Iterated Integral with variable substitution

I need to calculate the double integral of the function $f(x,y) = (x+y)^9(x-y)^9$: $\int_0^{1/2} \int_x^{1-x} (x+y)^9(x-y)^9 dydx$ I have a solution but I definitely arrived at it after a sloppy ...
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0answers
25 views

Maximum and Minimums (Calculus 3) [on hold]

a)Find the maximum and minimums of the function on the indicated closed and bounded region R. f(x,y) = xy-2x R is the triangular region with vertices (0,0) , (0,4) , and (4,0).\ b) Find the ...
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1answer
43 views

evaluating the double integral ∫∫ydxdy

The question is as follows: Use the transformation in Example 3 to evaluate the integral $$\iint_R y\ dx\ dy$$ where R is the region bounded by the x-axis and the parabolas $y^2 = 4-4x$ and $y^...
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0answers
24 views

Finding the velocity of a position vector

Let $\{\tilde{i}, \tilde{j}\}$ be the standard basis vectors for IR2. Define two paths in IR2 by $\tilde{v1}$(θ) = cosθ$\tilde{i}$ + sinθ$\tilde{j}$ $\tilde{v2}$(θ) = −sinθ$\tilde{i}$ + cosθ$\tilde{j}...
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1answer
46 views

Be $f:\mathbb{R}^{2}\to \mathbb{R}^{2}$ a continuous function and $g(x)=\int_0^1 \! f(x,y) \, \mathrm{d}y.$ Proves that g is continuous.

I don't see how to solve the following problem, I think that it's like a generalization of the fundamental theorem of calculus. Be $f:\mathbb{R}^{2}\to \mathbb{R}^{2}$ a continuous function and $g(...
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1answer
50 views

When does a sequence (or a series) of real-analytic functions converge to a real-analytic function?

It is well known that a sequence (or a series) of holomorphic functions converging uniformly converges to a holomorphic function. I would like to know under what condition a sequence (or a series) or ...
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1answer
8 views

Constrained optimization with multiple variables

I'm bringing a word problem/conceptual problem to the math stack exchange. Feel free to edit the title of this question if it does not reflect the following word problem/conceptual problem. What I ...
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2answers
33 views

Vector field and its components

If I have a cartesian vector field just denoted $\mathbf{E}=\mathbf{E}(x,y,z)$ (e.g. a electric field), does it mean: $$ \mathbf{E}(x,y,z)=x\mathbf{\hat{x}}+y\mathbf{\hat{y}}+z\mathbf{\hat{z}} \tag{1} ...
2
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1answer
69 views

Proving that a limit doesnt exist even if it exists

When I was trying to find a path that would prove that some limit doesn't exists, I was simply equaling the equation to a number and finding some expression. I will use some trivial limit, that can be ...
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2answers
45 views

Use lagrange multipliers to calculate the maximum and minimum

$f(x,y,z)=x^2y^2z^2$ constrained by $x^2+y^2+z^2=1$ $\nabla f_x$ $=$ $2xy^{2}z^{2}$, $\nabla f_y$ $=$ $2yx^{2}z^{2}$, $\nabla f_z$ $=$ $2zx^{2}y^{2}$ $\nabla g_x$ $=$ $2x$, $\nabla g_y$ $=$ $2y$, ...
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1answer
40 views

Work done by a Force Field( Green's Theorem)

Question: Compute the work done by a force Field $ F(x,y)=(2xe^y-x^2y-\frac{y^3}{3},x^2e^y+sin(y)) $ when a particle moves moves around the path describe by $ r(t)=(1+cos(t),sin(t)),0 \leq t \leq \pi ...
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0answers
23 views

Diagram of a multivaribale function

I have to draw the diagram of the function: $$(x^2+y^2)^{\frac{3}{2}}=x^2-y^2$$ I transformed it with polar coordinates to: $$r=\cos^2(\varphi)-\sin^2(\varphi)$$ with $r \ge 0$ and $\cos^2 \ge sin^2$....
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0answers
22 views

Line integral with differentials (cylindrical/spherical)

How can I write a line integral of a vector field with exact differentials in cylindrical and spherical coordinates? I know for in cartesian coordinates: $$ \mathbf{E}(x,y,z)=P\mathbf{\hat{x}}+Q\...
2
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1answer
31 views

Existence of absolute maxima and minima

In which of the following functions can be guaranteed the existence of absolute maxima and minima? a) $f(x,y,z)=x+y$ with $z\geq x^2+y^2+1$. b) $f(x,y)=\ln (x^2+y^2+1)$, with $x\geq 0$ and $y\geq 0$...
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0answers
12 views

Why in the definition of multiple integrals on subset $A\subset \mathbb{R}^n$ it is required that $A$ is measurable?

I'm new with the study of multiple integrals. I think I understood the topics of Peano–Jordan measure. A multiple integral is defined on a measurable (and limited) subset $A\subset \mathbb{R}^n$, ...
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28 views

Closest point to the origin from a Line

Let the curve $C\subset \mathbb{R}^3$ be the image of the map $\gamma:\mathbb{R}\to\mathbb{R}^3$, given by $\gamma(t)=(t,t^2,t^3)$. Let $R$ be the tangent line to $C$ at $(1,1,1)$. Find the point of $...
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22 views

Directional derivative of a differentable function

Is it always true, for a differentiable function $F:\mathbb{R}^N\rightarrow\mathbb{R}^M$, that its directional derivative along a direction $v\in\mathbb{R}^N$ is equal to the product $J_f\cdot v$, ...
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0answers
25 views

What are interesting functions in 2D that vary visually as compositionality increases?

I wanted to create a function that its shape was a function of the depth of the compositionality (on a fixed interval). For example consider some compositional function $$f(x_1, x_2) = g( g( g( h_1(...
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1answer
22 views

Why is multivariable continuous differentiability defined in terms of partial derivatives?

Both in my textbook and on Wikipedia, continuous differentiability of a function $f:\Bbb R^m \to \Bbb R^n$ is defined by the existence and continuity of all of the partial derivatives. Since there is ...
2
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1answer
22 views

Missing continuity condition in theorem?

I'm going through the proof that all partials continuous $\implies$ $f$ is differentiable. Here's what my book says: What I'm wondering about is how we can use the mean value theorem in step $2$. ...
1
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2answers
24 views

Differentiating $\int\cdots \int f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)~dx_1\cdots dx_n$

Differentiating:$$\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)\,dx_1 \cdots dx_n$$ with respect to $\theta$. The result is ...
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0answers
27 views

How to write/calculate an integral that covers $x\in\mathbb C$?

Consider some function $f(x)$ defined for $x\in\mathbb C$, and it is integrate-able as well. If I wanted to calculate the 'area' under the graph for $x\in\mathbb R$, I could use $\int_{-\infty}^\...
2
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1answer
38 views

Bounding the absolute error of the linear approximation by $|E|\le\frac{n^2M}{2}\|\mathbf h\|^2$

Let $f: D\subseteq \Bbb R^n \to \Bbb R$ be a $C^2$ function. I'm trying to show that the absolute value of the error of the first order Taylor approximation of $f(\mathbf x+\mathbf h)$ is bounded ...
1
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1answer
32 views

If for every $x\in\mathbb{R}^{3}$ the rank of the derivative $Df(x)$ is 2, prove that the image of $f$ is an open set.

I don't see how to solve the following problem, any suggestions? Let $f:\mathbb{R}^{3}\to \mathbb{R}^{2}$ such that $f\in C^{1}$. If for every $x\in\mathbb{R}^{3}$ the rank of the derivative $Df(...
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2answers
31 views

total derivative of function

Suppose one has a function: $G(x,y) = H(x,y) + L(x,y)$ Is it possible to evaluate the total derivative of $G$ with respect to $H$? That is, is it possible to compute, $\frac{d G}{d H}$ ?
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1answer
61 views

Geometric Significance of some features of the Exterior Algebra

I've been tinkering with differential forms for a while now, and I've had a few questions all rolled into one trying to understand them. The exterior derivative is quite natural to me - it looks just ...
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2answers
47 views

Partial and total derivative of a multivariable function

Given a function $f(x,y,t)$, is it correct to say $$\frac{d f}{d x} = \frac{\partial f}{\partial x} \text{ ?}$$
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1answer
21 views

Gradient of a maximum

How do you compute the gradient of a function that involves a maximum? For example, I have the function: $$ f(\vec{t}) = v(1-\exp(-\lambda\cdot \max(t_0,t_1)))$$ With $v$ and $\lambda$ constant, for ...
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3answers
84 views

A paradox in differential calculus

Say I have a function $f=f(x,y)$ where $x,y$ are independent variables. Now, it is given that $p=x+y$. It can be shown that, since $x,y$ are independent, we get $$\frac{\partial p}{\partial x}=\frac{...
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2answers
26 views

Why are the $r$ and $\theta$ unit vectors defined as such?

I refer to this derivation of the gradient in polar coordinates: http://www.math.jhu.edu/~js/Math202/polar.grad.chain.pdf I can understand all parts except why the unit gradient $$\hat{e_r}=\langle\...
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1answer
12 views

Intuitive way to understand Polar Coordinate Gradient

I am looking for an intuitive way to explain the "$1/r$" factor in the gradient in polar coordinates. For instance, if $g(x,y)=f(r,\theta)$, $$\nabla g=f_r\hat{e_r}+\frac 1rf_\theta\hat{e_\theta}$$ ...
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1answer
20 views

$\nabla g\cdot\hat{r}=\frac{\partial g}{\partial r}$ (polar coordinates)

$$\nabla g\cdot\hat{r}=\frac{\partial g}{\partial r}$$ I just want to check whether my understanding of $\nabla g\cdot\hat{r}=\frac{\partial g}{\partial r}$ is correct, where $\nabla g$ is in polar ...
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1answer
15 views

Determining Bounds to calculate mass

Let $E$ be the solid region defined by the inequalities $x \ge 0$, $0\le z \le \sqrt(x^2 + y^2)$, $x^2 + y^2 + z^2 \le 4$ Suppose that $E$ has mass density $\mu(x,y,z) = xz$. Calculate the ...
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2answers
43 views

Estimate $\int_{\|x\|\ge\delta}\frac1{\|x\|^{d+1}}\mathrm d x$ without spherical coordinates.

Is it possible to estimate the following Lebesgue integral ($\|\cdot\|$ is the 2-norm) $$\int_{\|x\|\ge\delta}\frac1{\|x\|^{d+1}}\mathrm d x, \, x\in\Bbb R^d$$ in terms of $\delta$ when $\delta\to 0$? ...
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2answers
56 views

How to show (x, y) = (0, 0) is the only solution to $4x^2 + 3y^2 + \cos(2x^2 + y^2) = 1$

I can't seem to think of/find a solution to this problem: Show that $(x, y) = (0, 0)$ is the only solution to $4{x}^{2} + 3{y}^2 + \cos(2x^{2}+y^{2}) = 1$ How would one go about proving this?
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0answers
21 views

Show that the vector path is regular?

Let $\{\tilde{i}, \tilde{j}, \tilde{k}\}$ be the standard basis of vectors for IR3. If the path $\tilde{x}$ : IR → IR3 is defined by $$\tilde{x}= \cos 4t\ \tilde{i}+ \sin 4t\ \tilde{j}+ 2t^2\ \tilde{k}...
2
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1answer
75 views

Integrating $e^{a\cos(\phi_1-\phi_2)+b\cos(\phi_1-\phi_3)+c\cos(\phi_2-\phi_3)}$ over $[0,2\pi]^3$

I am trying to integrate the following function. (it arises in channel modeling in wireless communications, Rayleigh random variables)..Any help is appreciated.Thanks $$\int_0^{2\pi}\int_0^{2\pi}\...
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4answers
164 views

How to show $ \lim_{(x,y) \to (0,0)} \frac{3x^2y^2}{x^4+y^4}=\frac{3}{2} $ using the $\epsilon$-$\delta$ notation.

I need to prove that: $$ \lim_{(x,y) \to (0,0)} \frac{3x^2y^2}{x^4+y^4}=\frac{3}{2} $$ using the $\epsilon$-$\delta$ notation. I have tried everything I could think of to make the expression into a ...
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1answer
65 views

Why is this function smooth on the coordinate axis

Consider the function $$f(x,y):=\sqrt{x^2+xy+y^3}, \quad x,y \geq 0.$$ It is claimed that this function is smooth except at the origin. I am wondering why this function is not smooth at (0,0) in the ...
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2answers
61 views

Understanding how to calculate surface area of parametrized surfaces

I am trying to follow a derivation for surface area of a parameterized surface and my book does not explain the reasoning behind different steps. I understand the derivation for surface area for a ...
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1answer
29 views

How to linearlize level curves at a saddle point

Let $f(x,y)$ be a real-valued function on a domain $D$ in $\mathbb{R}^2$, and let $(x_s, y_s)$ be a saddle point of $f(x,y)$ in $D$. That is to say, \begin{align} \frac{\partial f}{\partial x}(x_s, ...
2
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0answers
31 views

Equality of mixed partials proof

I'm trying to prove the equality of mixed partials. My book has a proof but it's only for functions $\Bbb R^2 \to \Bbb R$ (and then that can be extended to $\Bbb R^2\to \Bbb R^n$ by applying the ...
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25 views

how can I prove this property is the necessary and sufficient condition [closed]

enter image description here Here is related formula! please click the link!
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3answers
47 views

Find point on a line that is nearest to the origin

Can you help me with this exercise? Find the nearest point to the origin $(0,0,0)$ in the line given by the intersection of planes $x+y+z=2$ and $12x+3y+3z=12$. The intersection of the planes is ...
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0answers
15 views

Calculating Tangent Vectors

I`m trying to find unit tangent vector for each points on a curved line. I have the points in 3D space as x, y, z coordinates. For the sake of this example let's consider three points - P1(x1,y1,z1), ...
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0answers
16 views

The projection of the curvature vector onto tangent plane on Cone

Draw diagrams for cone ( with cone angle less than $360^{\circ}$) to show that the geodesics (generating ray and the warp around) have a projection of the curvature vector onto the tangent plane that ...