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
126 views

Set up integral in spherical coordinates outside cylinder but inside sphere

I have the equation of a cylinder and the equation of a sphere given: Cylinder: $x^2+y^2=4$ Sphere: $x^2+y^2+z^2=25$ I'm asked to set this up in cylindrical and spherical coordinates. Cylindrical ...
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2answers
413 views

A question on notation: What does $\nabla |\vec{a} \times \vec{r}|^n$ mean?

I sort of asked a version of this question before and it was unclear; try I will now to make an honest attempt to state everything clerly. I am trying to evaluate the following, namely $\nabla w = ...
0
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1answer
29 views

What space curves can this theorem describe?

We were given the following theorem in our Vector Calculus class: THM: For space curve $R$ which does not pass through the origin, and which has a second derivative, the following are equivalent: ...
2
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0answers
41 views

Spaces of the derivative in a direction

I have two question regarding the spaces where the first, and second, directional derivatives of a functional are. Let $\Omega \in \mathbb{R}^2$ an open subset. Let the functional: $$\phi =L^p ...
3
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1answer
244 views

Using Stokes' Theorem to evaluate $\displaystyle\int_{C}{(xyz)dx+(xy)dy+(x)dz}$

Let $C$ be the closed, piecewise smooth curve formed by traveling in straight lines between the points $(0,0,0),(2,1,5),(1,1,3)$ and back to the origin, in that order. Use Stokes' theorem to evaluate ...
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2answers
51 views

Is this function continuous? (vector function)

Assume you have $k$ vectors: $\{v_1,\dots,v_k\}$ in $\mathbb{R}^n$, and $\lambda\in\mathbb{R}^k$. Look at the function: $F\colon\mathbb{R}^k\rightarrow \mathbb{R}^n$ where ...
0
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1answer
42 views

Why do the limits of integration matter in a double integral?

Okay, I know that seems like a stupid question but I couldn't think of a better way to phrase it. I was trying to understand why iterated integrals involve "projecting" the domain onto one of the ...
0
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1answer
95 views

Show that $f$ is a linear map if $f$ is differentiable and its derivative is constant:

Show that if $f:ℝ^m→ℝ^n$ is a differentiable function whose derivative function $f′$ is a constant function and such that $f(0)=0$, then $f$ is a is a linear map. I am a little lost on this. I know ...
3
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0answers
27 views

Finding the Area of a Torus-like surface

I'm trying to find out the Area of the following surface: Let $C$ be the curve associated to a regular, simple path $\theta:[0,l]\rightarrow \Bbb R^2 $; also assume that ...
2
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1answer
26 views

A question regarding Surface Integrals and Stoke's Theorem

Let $G$ be an open set in $ \Bbb R^3$ and $F:G \rightarrow \Bbb R^3-{0}$ a vectorial field of class $C^1$. Suppose that $S$ is an open set, contained in $G$, whose non-empty boundary $\delta S$, is ...
1
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1answer
50 views

Calculating Flux through surface, stokes theorem, cant figure out parameterization of vector field

I am a tutor and trying to solve this for a student. I understand that Stokes's theorem allows us to compute the flux through the surface S, instead through the surface of the unit disk because they ...
0
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0answers
114 views

Finding the volume between a cone and a sphere

I have to find the volume between the sphere $x^2+y^2+z^2=1$ and below the cone $z=\sqrt{x^2+y^2}$ using Spherical Coordinates. Here is what I have so far: Transforming the cone part gives: ...
2
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1answer
51 views

Vector field with gradient and integral over curve

The problem is: Consider the vector field: $$\textbf{F}= 4x^3y^3 \,\textbf{i} + (1+3x^4y^2) \,\textbf{j}$$ a) Find a potential function $ϕ(x,y)$, i.e. a function $ϕ(x,y)$ such that $\nabla ϕ= ...
3
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1answer
93 views

Rank of the differential

Let $f:\mathbb R^n \to \mathbb R^n$ such that $f$ maps roots of a polynomial to its coefficients. Meaning: if $(x-x_1)(x-x_2)...(x-x_n)=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_n$ then $f\begin{pmatrix} x_1 ...
0
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2answers
41 views

Integrating exponential function with elliptic bounds

I am trying to integrate the following: $$\iint_R\exp\left(\frac{x^2}{4}+\frac{y^2}{16}\right)\:\mathrm{d}A$$ With the region $R$ having the bounds: $$\frac{x^2}{4}+\frac{y^2}{16}=3$$ ...
1
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0answers
27 views

Derivative with respect to r in terms of Cartesian coordinates?

I am trying to show that $$\sum_{j=1}^3 x_j \frac{\partial}{\partial x_j} = r \frac{\partial}{\partial r}.$$ I have $$\frac{\partial f}{\partial x_j} = \frac{\partial f}{\partial r} \frac{\partial ...
3
votes
2answers
109 views

What is difference between all of these derivatives?

In calculus II we were introduced to a bunch of new derivatives: the gradient, the derivative $D=\begin{bmatrix} \partial_{x_1} \\ \partial_{x_2} \\ \vdots \\ \partial_{x_n}\end{bmatrix}$, the ...
0
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1answer
59 views

Minimum where Hessian is positive semidefinite without continuous second derivatives?

I knew that, if function $f:A\to\mathbb{R}$ of class $C^2(A)$ in an open set $A\subset \mathbb{R}^n$ has a maximum, or respectively the minimum, in $x_0\in A$, then the Hessian matrix is positive ...
-1
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2answers
45 views

Find volumes using calculus?

The volume below by $z=\sqrt{x^2+y^2}$ and above by $x^2+y^2+z^2=1$. My Solution: Wrote the integral. Converted it into cylindrical coordinates. But keep getting $0$ as my answer. Can someone help ...
0
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1answer
44 views

Finding the hypervolume of a hypersphere

The question asked was: Find the volume of the hypersphere of equation x^2+y^2+z^2+w^2=a^2 using integration . I found the volume to be (1/2)(π^2)(a^4) using spherical coordinates but I need to also ...
2
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0answers
41 views

Determining solid region from bounds of triple integral

If you have an integral such as: $$\int_0^1\int_0^{2-x^2}\int_0^{2-x}f(x,y,z)dydzdx$$ How can you determine the equation for the solid region represented by the bounds of this triple integral? Does ...
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2answers
9 views

Generating points from bi variate normal mixture densities

I am asked to generate 200 and 1000 points from a bi-variate normal mixture densities. I am trying to understand the algorithm, not just the matlab code (I have to write it, not use an existing ...
4
votes
1answer
148 views

What is the meaning of $dA$ in double integrals?

What is the meaning of $dA$ in $\iint_E\dots dA$, where $E$ is a region in the $xy$ plane? In some integrals we use $dA=dx\,dy$, but in others $dA=\hat{k}\,dx\,dy$. (Here $\hat {k}$ is the unit ...
0
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1answer
32 views

A problem concerning chain rule and partial derivatives.

Let be $G : \mathbb{R}^3 \to \mathbb{R}$ a $C^{\infty}$ function. Suppose to indicate with $x,y,z$ the standard variables in $\mathbb{R}^3$, and another $C^{\infty}$ map $f: \mathbb{R}^3 \to ...
2
votes
2answers
85 views

Does the inverse function theorem fail for $\frac {\partial r}{\partial x}$

This is a follow-up to a question that I answered (though, not well enough). Why is it that $\frac {\partial r}{\partial x} = \cos(\theta) = \frac {\partial x}{\partial r} = \frac {\partial}{\partial ...
0
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1answer
27 views

Solving a line integral - splitting up in int multiple “segments”

I have the following formula: $$f = \oint\frac{ds}{C}$$ This integral happens over a (closed) circle with radius $r$, so normally the solution would be: $$\oint\frac{ds}{C} = \frac{2\pi r}{C}$$ ...
-1
votes
2answers
39 views

Mass of a half disc [closed]

A half disc of radius $R$ centered at the origin $(0,0)$, with $y>0$ and density $$\rho = \frac{\rho_oy}{\sqrt{x^2+y^2}}\qquad \mbox{for}\, y>0$$ where $\rho_o$ is a constant with units of ...
0
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0answers
16 views

Taylor coefficents of a function

I'm having some troubles for proving: Prove that in the Taylor Polynomial of f(x,y)=sin(xy) centred in (0,0) just the order 4k-2 coefficents are non zero. k={1,2,..n} I don't think induction is a ...
1
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1answer
192 views

How to derive differential volume element in terms of spherical coordinates in high-dimensional Euclidean spaces?

How to derive differential volume element in terms of spherical coordinates in high-dimensional Euclidean spaces (explicitly)? A derivation is here but its conclusions seems not right? The expected ...
2
votes
2answers
47 views

Divergence Theorem to calculate flux

Take the vector field given by: $F= (y^2+yz)i+(\sin(xz)+z^2)j+z^2k$ a) Calculate the divergence, $\operatorname{div}F$. b) Use the divergence theorem to calculate the flux $$\int_S F\cdot dA $$ ...
1
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3answers
93 views

When is this vector valued function pointing towards the origin?

"A fighter plane, which can shoot a laser beam straight ahead, travels along the path $\mathbf{r}(t) = \langle 5 - t, 21 - t^2, 3 -\frac{1}{27}t^3\rangle$. Show that there is precisely one ...
1
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1answer
39 views

Why is the “Normal Vector” normal?

I was trying to understand why the unit normal vector is normal to the direction of motion. Note that $\mathbf T(t) = \frac{\mathbf r'(t)}{||\mathbf r'(t)||}$ is the unit tangent vector for some ...
1
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1answer
44 views

The differential $\frac{\delta}{\delta w} w^T \Phi^T \Phi w = 2 \Phi^T \Phi w $

Let $w$ be a $n \times 1$ vector and $\Phi$ be a $n \times n$ matrix. Then the following differentiation rule holds: $\frac{\delta}{\delta w} w^T \Phi^T \Phi w = 2 \Phi^T \Phi w $ Is there some ...
3
votes
1answer
31 views

uniform continuity, differentials

Let $\{f_n\}_{n=1}^\infty$ be a sequence in $C^1(U)$ where $U \subset \mathbb{R}^d$ is open. Suppose $f_n \to f$ uniformly on compact subsets of $U$. Assume further that $df_n \to A$ in the same sense ...
0
votes
1answer
265 views

Integrating a solid using cartesian, cylindrical and spherical coordinates

The region $W$ is the cone shown below (see image). The angle at the vertex is $π/3$, and the top is flat and at a height of $7\sqrt{3}$. Write the limits of integration for $\int_W dV$ ...
4
votes
2answers
67 views

Convolution integral problem

In the process of solving a certain PDE, I've arrived at a convolution integral: $$\int_{\mathbb{R}^3} G(x-y) \nabla p(y) dy$$ where $x \in \mathbb{R}^3$, $G(z)=\frac{1}{\| z \|}$ and $p(z) = ...
1
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1answer
119 views

Example: Dirac function in two dimensions.

I am considering an equation of the form $$\int_{\mathbb{R}^{2}} f(x) \delta(s - x \cdot \theta)dx$$ where $\delta$ is a two-dimensional Dirac function. What does this evaluate to, exactly? I know ...
0
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0answers
45 views

Finding saddle points with some optimization

Consider a function that has multiple saddle points e.g. $$(x+y)(xy+xy^2)$$ We know for functions of two variables we can use the 2nd derivative test to locate saddle points $$Saddle ...
0
votes
1answer
68 views

Find the mass of a sphere with density given by $\rho(r,\theta,\phi)$

The density is given by $$\rho=\rho_0e^{-r/R}(1-\cos\theta)$$ Where $R$ is the radius of the sphere. I integrate as follows: So, first integration is $$\int_0^R \rho_0 e^{-\frac{r}{R}} (1-\cos ...
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0answers
24 views

Does It Make Sense to Take the Partial Derivative of a Directional Vector?

Let $\Omega$ be a domain in $\mathbb R^2$ and $C$ be a smooth curve wholly contained in $\Omega$. Moreover, $C$ is parametrized by $x(t), y(t)$. If $Q: \Omega \rightarrow \mathbb R$ has continuous ...
2
votes
0answers
77 views

Directional derivative (Vector)

Given $f:\mathbb{R}^2 \to \mathbb{R}^2$ is a map $f(x,y)=(u(x,y),v(x,y))$ and $\alpha=(\alpha_1,\alpha_2)$ is a point, then how does one show that $f$ is differentiable (or not) in the direction ...
1
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1answer
24 views

Evaluate Surface Integral over this triangular surface

When I solving the practice excericse problems at the end of the section, I stumbled upon this problem, which I have been trying to figure out how to compute the integral,but cant. Can someone please ...
1
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1answer
34 views

Finding the integral using divergence theorem without vector field

Use the divergence theorem to evaluate $$\iint_S (x^2 y^2+y^2 z^2+x^2 z^2)\ dS,$$ where $S$ is the entire surface of the sphere of unit radius, centered at the origin. How do I find this? ...
1
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1answer
28 views

Proving there exists a curve whose tangent vector $v$ satisfies $\nabla f \cdot v = 0$

Let $f:\mathbb{R}^3\to \mathbb{R}$ a $C^1$ function, $(x_0,y_0,z_0)\in \mathbb{R}^3$ such that $f(x_0,y_0,z_0)=0$ and $\nabla f(x_0,y_0,z_0)\neq 0$. Let $$S=\{(x,y,z) \ | \ f(x,y,z)=0\}$$ and ...
4
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0answers
53 views

The missing vector derivative operation

Generally, the differential operator $d$ takes differential $k$-forms to $(k+1)$-forms. In $\mathbb{R}^n$ (or more generally on an oriented Riemannian manifold), we can identify vector fields with ...
0
votes
0answers
40 views

Surface integral of the union of two lines in 3D space

Let $a(t) = \langle 0, 0, t\rangle$ and $b(t) = \langle\cos(t), \sin(t), t\rangle$. Let the union between $a(t)$ and $b(t)$ make the surface, $S$. Calculate the surface integral, $F \cdot dS$, where ...
0
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2answers
50 views

implicitly differentiating polar equations

For polar coordinates, we have the following equations. $x^2 + y^2 = r^2 $, $x= r \cos(\theta) $, and $y= r \sin(\theta)$. When I find $ \frac {\partial r}{\partial x}$, I have the following: ...
0
votes
1answer
51 views

Mass of Solid, Multivariable Calculus

Calculate the mass of the solid that lies above the surface $z= 0$, below the surface $z=y$, and inside the surface $x^2+y^2 = 4$ with the given density $yz$. I have switched to cylindrical ...
0
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1answer
24 views

Confused on how to find a tangent

Suppose $\operatorname{grad} f$ is tangent to the hyperbolas $xy = \mathrm{constant}$. I'm confused on how to find the tangent. I found the partial derivatives for $xy = c$, $\dfrac{df}{dx} = y$ ; ...
1
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1answer
13 views

Boundedness of multivariable polynomials

How can we prove multivariable polynomials are bounded on a closed set? the boundedness theorem is for single variable functions. Does an extension theorem exist? Thank you.