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
20 views

Find the area of a region using Green's Theorem.

Let $R$ be the region enclosed by the curve parameterized by $g(t) = (t^4 - t^2, t^6 - t^2)$, where $0 \leq t \leq 1.$ Find the area of $R$ using Green's Theorem. In order to use the green's theorem, ...
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2answers
45 views

Determining the equation of this 3D object

Does anyone know how I can determine the equation of the 3D object below? (Maybe there's a program that can do it?) I am looking for a formula to define this 3D object, but am having trouble finding ...
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0answers
13 views

Stadium billiard reflection angles

Given a boundary and a massless particle with constant velocity with a certain direction, a billiard consists of an experiment where the particle collides with the walls conserving its velocity ...
7
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1answer
494 views

Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?

Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem". I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
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2answers
34 views

Does being a local minimum imply a positive definite hessian?

If $p\in R^{m}$ is a local minimum of $F:R^{m}\rightarrow R$, then can we conclude that $\dfrac{\partial ^2F}{\partial x \partial x'}[p]$ is positive definite? I guess you guys answers have ...
4
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1answer
71 views

Gradient in polar coordinates

The gradient in the normal direction on a given contour of a function, expressed in polar coordinates as $u = u(r, \theta)$, can be calculated as: $$\frac{\partial u}{\partial n} = \frac{\partial ...
0
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1answer
452 views

determine the resultant speed of the plane

a plane flying due east at 200 km/h encounters a 40-km/h wind blowing in the north-east direction. the resultant velocity of the plane is the vector sum v = $v_1$ + $v_2$, where $v_1$ is the velocity ...
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2answers
35 views

Taking derivatives of the implied function - from the implicit function theorem,

I showed that the relation $$f(x,y)=e^x - e^y + xy = 0$$ defines near (0,0) an implicit function y=$\phi (x)$, since the $1x1$ block, $\frac{df}{dy}$, evaluated at (0,0) gives -1, which is non-zero - ...
2
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1answer
29 views

introduction to potential theory in $\mathbb{R}^3$ [on hold]

A differentiable function $g: \mathbb{R}^3 \to \mathbb{R}$ is said to be harmonic in a subset $B \subset \mathbb{R}^3$ if $\Delta^2 g = 0$ for all $p \in B$. Let $M \subset \mathbb{R}^3$ be a bounded ...
1
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1answer
59 views

Finding points that satisfy $f(a) = \sup f(x)$

Choose positive real numbers $\alpha_1,\ldots,\alpha_n$, $n$ such that $\sum_{i=1}^n \alpha_i = 1$ and let $$f: [0,\infty)^n \to \mathbb R$$ $$x=(x_1, \ldots, x_n) \mapsto x_1^{\alpha_1} \cdots ...
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1answer
111 views

Exponential of a matrix and related derivative

I have $ X \in M(n,\mathbb R) $ to be fixed. I define $ g(t) = \det(e^{tX}) $ Then the author proceeds as follows: $ g'(s) = \frac {d}{dt} g(s+t) $ = $ \frac {d}{dt} \det(e^{(s+t)X}) |_{t=0} $ = $ ...
3
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1answer
26 views

Find the area of a subset of $\mathbb{R}^3$ given by an implicit relation.

Let x, y, z be real numbers and let $A = \begin{bmatrix} 1&x&x^{2} \\ 1&y&y^{2} \\ 1&z&z^{2} \end{bmatrix} $ Let S be the subset of $\mathbf{R}^{3}$ given by $S = \{ ...
0
votes
2answers
29 views

derivative of domain of integration

Suppose $f(x,y)$ is a function $\mathbb{R}^2\times \mathbb{R}^2 \to\mathbb{R}$ and $\Omega(x)$ is a family of compact regions of the plane whose boundary curve $\gamma(s,x)$ varies smoothly in $x$. I ...
1
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2answers
132 views

Equality of mixed partial derivatives

Is the following statement $$\frac{\partial^2 f}{\partial x \, \partial y}=\frac{\partial^2 f}{\partial y \, \partial x}$$ always true? If not what are the conditions for this to be true?
4
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2answers
30 views

Integration by Change of Variable

By using change of variable, $$x+y=(\surd2)u \text { and } y-x=(\surd2)v$$ Evaluate $$I=\iint(y-x)^2e^{-(x+y)^2}dv\,du$$ with $R$ bounded by $x=0,y=0,x+y=1$ After changing of variable, I get ...
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2answers
28 views

Help me to understand Line integral problem solution

I as thinking as to why theta has been taken from 0 to 2pi .it is not obvious from picture . Also Can it be done using stokes .Thanks
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0answers
50 views

Seperating single integral into an double integral.

Please refer to : How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$ The answer by @Venus. What is the procedure in converting that ...
1
vote
1answer
56 views

Operator $\nabla$

My teacher told me to solve some physics problem where I need to find $\rho(r)$ by using: $$\frac{\rho(r).dV}{\varepsilon_0}=\vec E(r+dr)S(r+dr)-\vec E(r)S(r)$$ where $dV$ is a small hollow spherical ...
0
votes
1answer
77 views

Cauchy-Riemann equations and harmonic functions

I have the following question which I am supposed to use the Cauchy-Riemann equations to prove: Let u and v have continuous second derivatives satisfying: $$\frac{\delta u}{\delta x} = \frac{\delta ...
2
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2answers
1k views

Finding a vector parametric equation given P and Q equations?

Find a vector parametric equation $r⃗(t)$ for the line through the points $P=(3,5,4)$ and $Q=(1,4,7)$ for each of the given conditions on the parameter $t$. If $r⃗ (0)=(3,5,4)$ and $r⃗ (7)=(1,4,7)$, ...
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1answer
22 views

Tangent Vector at the point $(1,2,11)$ whose projection onto the $xy$-plane is parallel to vector $1/\sqrt{10}i+3/\sqrt{10}j$.

$$f(x,y) = x^3y^2 + 3x + 2y$$ The gradient of $f$ at the point $(1,2)$ is $15{\bf i} + 6{\bf j}$. ${\bf u} = \dfrac{1}{\sqrt{10}}{\bf i} + \dfrac{3}{\sqrt{10}}{\bf j}$ The Directional Derivative ...
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3answers
247 views

parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
2
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0answers
64 views

How show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$?

Question: I want to show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$. Issue: I know how to prove this via the epsilon-delta way. I ...
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0answers
15 views

Enlarge a 2d plot of f(x,y) while keeping the moiré patterns at the same size

I have some function $z=f(x,y)$ with $x,y,z$ real numbers. I wrote a script that generates a $600\times 600$ pixels plot of this function for, say, $0 < x < 1$ and $0 < y<1$, by mapping z ...
3
votes
2answers
27 views

Optimization - find the dimensions of a box as functions of volume - minimal surface area

Had a basic calculus course exam today. This was one of the problems: We have a rectangular box of a given volume V. Present the width, height, and length of the box as functions of V so that the box ...
1
vote
1answer
55 views

Question about milnor's proof of hairy ball theorem

Here is a link about the proof: http://people.ucsc.edu/~lewis/Math208/hairyball.pdf My question is: after lemma 2, Milnor takes the region A to be the region between two concentric spheres. Why can't ...
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1answer
53 views

Calculate surface area of a F using the surface integral

Task Given: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid (x,y) \in W,z=f(x,y)\}$$ Calculate the surface area using the surface integral: $i) \; f(x,y) := x+y \;\; and \;\; W := [12,31] \times ...
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2answers
277 views

Constructing a Cone and its Normal Vectors in Spherical Coordinates

I am attempting to construct a right circular cone of maximum radius $R$ and angle $\theta$ in spherical coordinates, then find the normal vector of the surface of this cone at all points. Here's what ...
2
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1answer
29 views

Using Lagrange multipliers to find the extrema of $f(x,y) = e^{2xy}$ subject to $x^2+y^2 = 16$

Find the maximum and minimum values of $f = e^{2xy}$ with respect to $x^2+y^2 = 16$. Using Lagrange multipliers, $\nabla f = \lambda\nabla g$. Therefore, the constraints are the following: ...
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2answers
35 views

How do I find this distance?

Find the minimum and maximum distances between the ellipse $x^2+xy+2y^2 = 1$ and the origin. This is what I've attempted so far: Maximize $x^2+y^2+z^2$ with respect to $x^2+xy+2y^2 = 1$. Using ...
3
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2answers
262 views

General solution for the system of PDEs from the curl of a vector field equaling another

In my vector calculus class, when we were introduced to the curl operator the professor gave us this example: Is it possible to find a vector field $\mathbf{G}$ such that $$\mathbf{F} = \nabla ...
4
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1answer
40 views

Can the Heat Equation be Averaged Over a Region?

I am doing a project for my partial differential equations class in which I am motivating the definition of a weak solution. To get started, I assumed that $T$ was a solution to $\nabla^2 T = \partial ...
1
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1answer
25 views

Construct a procedure which determines the location of the shadow of a rectangluar box.

I drew a 3d rectangular box on a coordinate plan consisting of x, y, and z. A procedure is to be created that will determine the location of the shadow of the box on one of the coordinate planes. I ...
3
votes
2answers
69 views

Integration of the vector field $\mathbf {F } (x,y)=\frac{y}{x^2+y^2}i-\frac{x}{x^2+y^2}j $ over two ellipses

Let $\mathbf{F}$ be a vector field defined on $\mathbb R^2 \setminus\{(0,0)\}$ by $$\mathbf {F } (x,y)=\frac{y}{x^2+y^2}i-\frac{x}{x^2+y^2}j $$ Let $\gamma,\alpha:[0,1]\to\mathbb R^2$ be defined by ...
0
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0answers
29 views

How to calculate the volume of a tetrahedron?

Suppose that $$ I=\iiint_{V}f(x,y,z)dxdydz $$ where $f(x,y,z)$ is a continuous function, $V$ is a tetrahedron whose vertices are $P(2,2,0), A(-2,0,0), B(0,0,2)$ and $C(1,1,3)$. I want to ask you how ...
0
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1answer
28 views

Calculate surface area of a sphere using the surface integral

Given a sphere with: $$F := \{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 = 1, x\le0\}$$ $$ \Rightarrow r = 1, \varphi = [\frac{\pi}{2}, \frac{3\pi}{2}], \theta = [0, \pi] $$ My Task is to calculate ...
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0answers
22 views

Geometric position of gradient on the surface in $\mathbb{R}^3$ and orthogonality to tangent of level curve

Given a function $f(x, y)\in C^1(\mathbb{R}^2)$ and its gradient $\nabla f(x, y) =(\frac{\partial f(x, y)}{\partial x}, \frac{\partial f(x, y)}{\partial y})$ which forms a vector field where each ...
2
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1answer
26 views

Calculate surface integral

I need some help with the following: Given $$f(x,y,z)=\left( \frac{-x}{(x^2+y^2+z^2)^{\frac{3}{2}}}, \frac{-y}{(x^2+y^2+z^2)^{\frac32}}, \frac{-z}{(x^2+y^2+z^2)^{\frac32}} \right),$$ calculate the ...
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1answer
113 views

What is the effect of axis rotation on functions defined on $\mathbb{R}^{2}$

I haven't studied multivariable calculus yet but I have a question that bothers me. Let $F$ be a function $\mathbb{R}^2 \to \mathbb{R}$. Imagine that we rotate the co-ordinate axis by an angle ...
4
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1answer
33 views

Independence of function and its derivative in calculus of variations

It's common to see in calculus of variation that the integrand $f$ of functional $F[y]=\int f(y,y',x)dx$ is a function of $y,y'$ and $x$. Why do we regard the derivative $y'$ as an independent ...
0
votes
1answer
36 views

Volume bounded by two solids

Can somebody help me get started in the right direction for this question involving volume? The question is "Find the volume of the solid region inside the hemisphere $x^2 + y^2 + z^2 =6, z<0$ but ...
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0answers
37 views

To find the Maximum and minimum value of f over square

Given function $f = (x+y)^2 - (x+y) +1$ .I have to find maximum and value of $f$ over square with unit side in first octant in xy-plane. I calculated $f_x $ and $f_y $ both came out to be ...
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1answer
21 views

How can you find the distance between two skew planes?

I understand that there is a unique line perpendicular to both planes and the length of that line is the distance between the planes but how would I go about finding the what the equation of that line ...
5
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1answer
190 views

Riemann Integrability in $\Bbb R^2$

Define the General Subdivision $S$ of a rectangle $R$ in $\Bbb R^2$ as a collection $E_1,...,E_k$ of Jordan regions such that none of them has interior points in common, and: $$R \subset ...
0
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1answer
33 views

Partial derivative is bounded

Let $f(t,z)$ be a bounded (say by a constant $M$) continuous function on $\mathbb{R}_t \times \mathcal{U}$ where $\mathcal{U}$ is an open neighborhood of $0 \in \mathbb{C}_z$. Moreover, for each fixed ...
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vote
1answer
346 views

Why gradient vector is perpendicular to the plane

I know what gradient vector or $∇F$ is and I know how to prove it is orthogonal to the surface (using calculation - not intuitive). In a particular case, in which we have a three variable function, I ...
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3answers
38 views

Does the method for finding the intersection of 2 single variable functions work for multivariable functions?

I have $2$ multivariable functions $Q(x,y)$ and $P(x,y)$, I was wondering if finding the point of intersection between these 2 functions is as easy as making $Q(x,y) = P(x,y)$ as you would do for most ...
3
votes
1answer
20 views

Chain rule notation for composite functions

Suppose I have a function $ f(x, y, g(x, y)) $ How would I express $ \frac{\partial f}{\partial x} $? Using the chain rule, you'd naturally come up with $ \frac{\partial f}{\partial x} + ...
0
votes
2answers
37 views

Finding the mass of a cone using triple integral

I have a density $\rho(x,y,z) = 3-z$ and have converted my given information to form a triple integral equation for finding the volume of my cone in cylindrical coordinates and have found the volume ...
1
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0answers
25 views

Finding the normal vector of a surface (Flux of a vector field n*dS expression)

This problem is practice for a final exam. Let S be the closed surface whose bottom face B is the unit disc in the $xy$-plane and whose upper surface U is the paraboloid $ z = 1 − x^2 − y^2 , z \geq ...