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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2answers
17 views

How can I find the limits of this iterated polar integration?

How can compute the area of the triangle whose corners are at the origin, (1,0) and (1,1). I solved this with r integral first but I could not find the correct limits for theta integral first order. ...
0
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1answer
27 views

How to integrate an equation with multiple non-independent variables

I'm a little lost with this particular equation, I have three variables which need to be integrated and can't quite wrap my mind to get the correct result. I have this: $$ \frac{dH}{dt}=8\pi ...
1
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1answer
61 views

Solving $\int_0^2\int_{y/2}^1 ye^{-x^3}\,dx\,dy$ [on hold]

So I have to solve the integral above but I wasn't really sure how to start? I know it is integrable since it is an integral over a nice boundary and that I can solve it using iterated integrals but ...
0
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1answer
28 views

If $h:\mathbb{R}^2 \to \mathbb{R}$ is some function, and $g(r,\theta)=(r\cos{\theta},r\sin{\theta})$, compute the matrix $Dh_{g(r,\theta)}$.

The question says to express $Dh_{g(r,\theta)}$ only in terms of $r$, $\theta$, and the two entries of $D(h\circ G)_{(r,\theta)}$, but I'm not really sure how. Could someone point me in the right ...
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1answer
40 views

Dude with taylor polynomial

Good night, i'm working with an problem of polynomial taylor, but i have a problem with the residue. Get a quadratic approximation $f\left(x,y\right)=\sin\left(x\right)\sin\left(y\right)$ near the ...
1
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1answer
23 views

find value or prove limit doesn't exist.

Given: Find or prove it doesn't exist: .... My attempts thus far include: I can show that doesn't exist using y=kx and showing path dependancy, but dunno if it's enough to prove that ...
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2answers
40 views

Polar equation of the curve y = sinx

I am looking for the polar equation of the following curve given in Cartesian Coordinates. y = sinx Any kind of hint or help is appreciated.
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1answer
9 views

How do I convert a integral over a region R into a double integral using Green's Theorem?

I have been given an integral ∫(xdx+2xdy) over the region R defined as the region bounded above by y=ex-x+1 and below by y=e^x, and I have been asked to convert it to a double integral using Green's ...
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0answers
16 views

Is differentiation with respect to a vector always defined componentwise?

When one takes the derivative of a function $f$ along the direction of some vector $\mathbf{v}$, i.e. the directional derivative of $f$ along $\mathbf{v}$ this operation is defined componentwise, i.e. ...
1
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1answer
39 views

How can I use inverse/implicit function theorem to find a function and its inverse?

My task is this: Let $f:\mathbb{R}^3 \to \mathbb{R}$ be the function $f(x,y,z) = xy^2e^z + z$. Show that there exists a function $g(x,y)$ defined around $\textbf{x} =(-1, 2)$ s.t. $g(\textbf{x}) = ...
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1answer
28 views

How do i show an iteration using Newtons method in $\mathbb{R}^m$?

My task is this: Let $\textbf{F}:\mathbb{R}^m \to \mathbb{R}^m$. A fixpoint for $\textbf{F}$ is the same as a zero for $\textbf{G}(\textbf{x}) = \textbf{F}(\textbf{x}) - \textbf{x}$. Show that when ...
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0answers
9 views

Concave or convex function defined on convex set.

I have a question regarding the definition of concave and convex functions for many variables. Both are defined for some convex set. I am wondering what happens for nonconvex sets. If anyone can help ...
2
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0answers
19 views

Maximum of a Line Integral of the Vector Field

The problem: Let $S$ be the graph of $z=\sqrt{1-x^2-y^2/2}$. Let $F=<x+y,xy,sin(e^x)>$ be a vector field. To which level curve does the line integral of the vector field attains maximum? How do ...
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1answer
24 views

Finding the Gradient of a Vector Function by its Components

In Multivariable Calculus, we can easily find the gradient of a scalar function (producing a scalar field) $f : \mathbb{R^n} \to \mathbb{R}$, and the gradient function would produce a vector field. ...
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1answer
65 views

Find the mass of the half circle

Find the mass of the half circle that is defined by $x^{2} + y^{2} \le 4$ $(y \le 0)$ if the density at point $(x,y)$ is proportional to its squared distance from the point $(0, -2)$ and the density ...
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0answers
19 views

integral of partial derivative for continuity equation

Task in question is deriving velocity parallel to z component in continuity equation. $$ \frac{\partial n}{\partial t} + \left(\nabla \cdot n \textbf{u}\right) = 0 $$ $$ \frac{\partial n}{\partial t} ...
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2answers
19 views

Find all Points on the Surface at which the Tangent is Parallel to the Plane

The problem: Find all points on the surface $z=x^3+xy^2$ at which the tangent plane is parallel to the plane $2x+2y+z=0$ So I established $f(x,y,z)=x^3+xy^2-z$ and the normal vector determined from ...
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3answers
40 views

Implicit Differentiation - What am I doing wrong?

I need to find $y'$for the following equation: $$ e^{\frac{x}{y}} = x-y $$ Before differentiating I decided to perform a quick rewrite: $$ \begin{align*} e^{\frac{x}{y}} &= x-y \newline ...
0
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1answer
31 views

How to prove the following inequality using Lagrangian multipliers?

Find the maximum and minimum values of the function $f(x,y,z)=(xyz)^2$ where $(x,y,z)$ is on the sphere $x^2+y^2+z^2=r^2$. Then show using above that $(abc)^{1/3} \leq (a+b+c)/3$. For non-negative ...
2
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1answer
49 views

Calculate this Triple integral!

They ask me find the following: W is the solid bounded by the limited right circular cylinder: $$ x^2+y^2=1$$ and the planes: $$z=0, z=4$$ must calculate: $$\iiint_W z\frac{e^{2x^2+2y^2}}{2} ...
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2answers
23 views

How do i show the existence of two secquences that converges to a border point?

My task is this: Show that if $\textbf{c}$ is a border point for $A\subset \mathbb{R}^m$, then there exists two sequences $\{\textbf{x}_n\}$ and $\{\textbf{y}_n\}$ that converges to $\textbf{c}$ such ...
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1answer
18 views

Defining a domain of cardioid region in terms of polar coordinates.

Consider the region contained inside both the cardioid $r=1+\cos\theta$ and outside the circle $r=3\cos\theta$, where $r$ and $\theta$ are polar coordinates. So weirdly enough I know how to calculate ...
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0answers
26 views

Higher Order Derivative Tests in Multiple Dimensions

To evaluate the minima, maxima, and saddle points of a real function of 2 variables, we use the second derivative test after evaluating the critical points to identify the type of extrema they are. ...
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0answers
9 views

Mixed derivatives of coordinates from two representations related by an orthogonal transformation

Given two orthonormal vector representations $\overline{Y}$ and $\overline{Q}$ of an $\mathbb R^n$ space that are related by an orthogonal transformation $\overline{Q} = ...
2
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0answers
57 views

Graphing the surface $z = xy$

Let the surface $S \subset \mathbb{R}^3$ be the graph of the function $f:\mathbb{R}^2 \to \mathbb{R}, f (x, y) = xy$. Let $U$ be the portion of $S$ for which $x^2 + y^2 ≤ 2$ and let $C$ be the ...
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2answers
18 views

Find the part of the solid $U$ that is inside the cone

I have an exercise in the book that says: Let $U$ be a solid that is defined by $x^2 + y^2 + z^2 \le 1$. Find the part of the solid $U$ that is inside the cone $z = \sqrt{\frac{x^2 + y^2}{3}}$. As ...
0
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1answer
12 views

Solving proportions with multiple variables.

If $3$ bulldozers can fill $5$ dump trucks with dirt in $30$ min, how long would it take $5$ bulldozers to fill $7$ dump trucks?
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1answer
58 views

Find the triple integral $\iiint_{\mathrm{T}}x^{4}\, dxdydz$

Find the triple integral $\iiint_{\mathrm{T}}x^{4}\, dxdydz$ where $T$ is bounded by $x^{2} + y^{2}+2y = 0$ and $z = 2x, z = 0$ planes. My attempt at the solution: $0\le z\le 2x, -2\le y\le 0, ...
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0answers
30 views

Uniform current in cylinder and straight wire causing same magnetic field?

The tridimensional version of the Biot-Savart law says that the magnetic field generated at the point $\boldsymbol{r}\in\mathbb{R}^3$ by a tridimensional distribution of current defined by the current ...
2
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0answers
28 views

Matrix Calculus and Linear Transformations

I'm working on making the jump from differentiating real valued functions ($f: \mathbb{R}^n \rightarrow \mathbb{R}$) and vector valued functions ($g: \mathbb{R}^n \rightarrow \mathbb{R}^m$) to matrix ...
3
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0answers
71 views

Sketching the surface $x^2+y^2+4z^2 = 1$

Let the surface $S \subset \mathbb{R}^3$ be the solutions of the equation $g(x, y, z)$ $ = 1$ where $g(x,y,z)=x^2 +y^2 +4z^2$. Let $U$ be the finite region of S satisfying $z > 0$ and let $C$ be ...
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1answer
14 views

Where is the error in the use of polar coordinates to find the volume of a cone.

Let the cone be opening along the $z$ axis and be of height $h$ and radius $a$. Then the region of integration would be $\theta \in [0,2\pi],~ r\in[0,a],~ z \in[x^2+y^2=z^2,h]=[r,h]$ but this does ...
2
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1answer
67 views

Does anyone know a book on sketching surfaces?

Is anyone aware of books or sources dedicated to sketching surfaces? Sort of like Forst's An Elementary Treatise on Curve Tracing, but on surfaces; or a book that has a fair few chapters dedicated to ...
1
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1answer
31 views

Surface of sphere above/below ellipse

I am struggling with the following problem: Find the surface area of $x^2+y^2+z^2=a^2$ enclosed by the cylinder $\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1$ $(a>b>0)$. The solution of the problem is ...
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0answers
42 views

Find the point where this function is not locally invertible

Define $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ by $f(x,y) = (x-y, xy)$. I have to find the points where $f$ is not locally invertible. The Jacobian is nonsingular at all points NOT of the form (x, ...
0
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1answer
21 views

Calculus: Applying coordinate changes to a simple PDE

The problem: Let $U(x,y)=V(t,s)$ where $x=t$ and $s=x\cos\alpha+y\sin\alpha$ and $\alpha$ is a constant in the interval $(0, \pi/2)$. Assume that $V_{tt}(t,s)+2\cos\alpha V_{ts}(t,s)-3e^t\sin\alpha ...
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1answer
32 views

Need help in understand how partial derivatives are taken in ML course from Coursera

I did not study multivariable calc, but I need to understand how the following derivatives are taken. I will really really appreciate your help! I took it from Coursera ML course. Thanks
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0answers
8 views

Closed-form solution for system of equations for finding a critical point

I am trying to find a critical point of a function $\mathbb{R}^d \to \mathbb{R}$ by setting its gradient to zero. I would like to solve the follwoing system of equations. $$\frac{1}{1 - \sum_{j=1}^d ...
2
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0answers
22 views

Vector Calculus: solution to Poisson equation

This is problem 8.4.17. from Marsden Vector Calculus book. Let $\rho$ be a continuous function which vanishes outside a 3D region $W$. Define ...
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0answers
22 views

Can we derive a Taylor formula for real-valued Fréchet differentiable functions on a normed space?

Using the Lagrange form for the remainder, Taylor's theorem can be stated as follows: Let $I\subseteq\mathbb R$ be an interval, $f\in C^{n+1}(I)$ for some $n\in\mathbb N_0$ and $s,t\in I$ ...
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0answers
20 views

Gradient of function f(x,y,z)=h

I have function $f(x,y,z)=h$. Function $f$ is linear and I have three points $A$, $B$, $C$ given by $x,y,z$ and $h$. How can I compute gradient of $f$ ? According to further use in the article I am ...
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0answers
39 views

Global extremum when the constraint is not compact?

When the constraint is compact, the function must have both a global maximum and a global minimum somewhere in the constraint. However, if the constraint is not compact, the global extremum may not ...
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1answer
26 views

Compute the Flux of the vector field

The vector field is this $F(x,y,z)=\left\langle z^2-y^2e^z,z\ln(1-2x^2),3\right\rangle$. S is a portion of the graph $z=5-x^2-y^2$ which sits above the plane $z=0$, and orientation upwards. Compute ...
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1answer
28 views

Calculate Surface Integral using Divergence

Here's the problem: Let $G(x,y,z)= \langle x+\cos z,y+y\sin x,z+\cos y-z\sin x\rangle$. Compute the surface integral $\int\int_SG\cdot ds$ where $S$ is the boundary of the solid which is inside the ...
2
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1answer
44 views

Finding the Gradient of a Tensor Field

Finding the Gradient of a Scalar Field I understand that you can find the gradient of a scalar field, in an arbitrary number of dimensions like so : $$grad(f) = \vec{\nabla}f = ...
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0answers
23 views

How to make the standard change-of-variables in the plane-parallel radiative transfer equation?

This is a basic technique used frequently in going from the general coordinate-free radiative transfer equation (RTE) to the RTE formulated for the plane-parallel atmosphere geometry (see Liou, ...
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1answer
50 views

Classifying the stationary points

Question: Find and classify the stationary points for the following function $$f(x,y) = (\frac{3}{2x}-\frac{x}{2}-y)^2 + x^2 +y^2$$ My attempt: I have found the stationary points for this function ...
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2answers
38 views

multivariable calculus exercise in the weak formulation of the Navier Stokes Equations

The following is an excerpt from Robinson's An introduction to the classical theory of the Navier–Stokes equations. Here (1.1): Using integration by parts I can get the $(\nabla u,\nabla\varphi)$ ...
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2answers
34 views

Analysis of $(y-x)^2 =x^3 $

I was doing some tasks in integral application and came across this one: Calculate the surface area bounded by $(y-x)^2 = x^3$ and line $x=2$ I started doing this the usual way, when I realized ...
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0answers
46 views

Please help… is this a convex function?

Kindly help me. What can we say about the function $f$ shown in below? is it convex or non-convex over the variables $x_1, x_2,.., x_{n+1}, y_1,y_2$? \begin{align} f(x_1, x_2,.., x_{n+1}, ...