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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Coordinate method for PDE

Solving the PDE $au_x+bu_y+cu=0$ The PDE is transformed by the coordinate method via, $\begin{cases}x'=ax+by\\y'=bx-ay\\\end{cases}$. What I don't understand is how should I know I have to pick ...
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60 views

Finding the volume of a 15-dimensional unit ball [using elementary multivariable calculus]?

It wouldn't surprise me if what I am asking is impossible (knowing my professor), but is there a way to find the volume of a 15-dimensional unit ball using elementary multivariable calculus? How ...
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60 views

Tricky Surface Parametrization

I am to parametrize the surface given by the ellipse $$9(z-1)^2 + x^2 = 1$$ in the $xz$-plane and rotated about the $x$-axis. I then have to find the volume of the region enclosed. The concept of ...
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123 views

Cauchy's Theorem and Cauchy's formula

I came across the following problem in our last midterm exam. I am completely stuck as to how to begin the solution: If $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), then $|f|$ has no ...
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2answers
54 views

$\oint_C{f\nabla f}\cdot d\mathbf{r}=0$ proof

I recently came across this statement which I was told was true. $$\oint_C{f\nabla f}\cdot d\mathbf{r}=0$$ Can anyone provide a proof of why this is the case? Is there any way to show that $$f\nabla ...
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26 views

Integrate over triangle

Integrate $f(x,y) = (x+y+1)^{-2}$ over the triangle with vertices $(0,0), (4,0), (0,8)$. I think you have to split up the triangle into different equations: $x=0, y=0, y=2x-8$. But I'm not sure what ...
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52 views

Calculating mean vector of a multivariate distribution

I have a question concerning calculating the mean vector (vector of expected values) of a general multivariate distribution. I try to obtain the mean vector by doing a vector integration and I ...
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189 views

Double Integral - Sketch region and evaluate

Sketch the region of integration and evaluate the integral: $$\int_1^2 \int_y^{y^2} dx \, dy$$ I understand how to take the integral, but the region of integration seems like it has no bounds. Like ...
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59 views

Differential equation have no particular solution for initial condition

Does the following diferential equation have a particular solution? $y'=(y-1)(y+1), y(0)=1$ The general solution is: $\frac{1-e^{2x+2C}}{1+e^{2x+2C}}$ But then: $\frac{1-e^{2C}}{1+e^{2C}}=1$ $C$ ...
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1answer
45 views

Tangent Plane w/ a Minus Sign?

The tangent plane to a surface $u = f(x,y)$ is given by $$z = f(x_0,y_0) + \tfrac{\partial f}{\partial x}|_{(x_0,y_0)}(x-x_0)+\tfrac{\partial f}{\partial y}|_{(x_0,y_0)}(y-y_0).$$ So why does a book ...
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114 views

Is $f$ differentiable at $(x,y)$?

I am working on a practice question, and I am not sure if what I have done would be considered, 'complete justification'. I would greatly appreciate some feedback or helpful advice on how it could be ...
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130 views

Show that the intersection of a plane…

Show that the intersection of the plane $z = 2y$ with the elliptic cylinder $\frac{x^2}{5} + y^2 = 1$ is a circle. Find the radius and center of this circle. Hint: How can one describe a circle in ...
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43 views

Find the parametric equations of the line of intersection…

Find the parametric equations of the line of intersection of the planes x - z = 1 and x + 2y + 3z = 1. I'm assuming it's something to do with cross product? Here's what I've set up: x y z 1 ...
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28 views

multivariable calculus question norm

Given vector space C([a,b],$ \mathbb{R} $) of continuous functions of [a,b] in $ \mathbb{R}. $ Prove that the function $ \left \| f \right \|_{1}=\int_{a}^{b}\left | f(t) \right |dt $ is a norm. Also ...
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325 views

The “second derivative test” for $f(x,y)$

I'm currently taking multivariable calculus, and I'm familiar with the second partial derivative test. That is, the formula $D(a, b) = f_{xx}(a,b)f_{yy}(a, b) - (f_{xy}(a, b))^2$ to determine the ...
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2answers
61 views

center mass of the solid

Find the center mass of the solid bounded by planes $x+y+z=1,x=0,y=0$ and $z=0$, assuming a mass density of $$\rho(x,y,z) = 10 \sqrt{z}.$$ I could not set up the integral!
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106 views

How to calculate volume of a solid under a given surface with double intergrals?

How can I calculate the volume of the solid under the surface $z = 6x + 4y + 7$ and above the plane $z = 0$ over a given rectangle $R = \{ (x, y): -4 \leq x \leq 1, 1 \leq y \leq 4 \}$? I know I have ...
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1answer
379 views

The motion of a solid object

The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density at the point and occupies a region W, then the ...
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1answer
95 views

Given a set of data points, how to use gradient descent to find the minimum in the function that passes from those data points?

I have a function with n parameters. I don't know the formula of the function but I can generate as many data points as I want using the function that I have. My question is, how can I find the set of ...
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Why this $C^1$ function is onto?

Let $f:\mathbb R^n\to \mathbb R^m $ is class $C^1$ Also $f^{-1}(B)$ is bounded whenever $B$ is bounded and $\nabla f_i(x)$ are linearly independent for each $x$. Then $f$ is onto. Why? I have no ...
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1answer
29 views

Taking the Double integral?

Let D be the (rotated) square in R^2 with vertices (+/- 1,0) and (0,+/-1). Let f(x,y) = 5, g(x,y) = xy, and h(x,y)= y^2. Compute the double integrals of f(x,y), g(x,y), and h(x,y) This question is ...
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22 views

How do you know when the derivative is a matrix or a sum?

For the derivative of a function, sometimes I see the derivative written as a matrix of the partial derivatives, and sometimes I see the derivative written as a sum of the partial derivatives. When do ...
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1answer
116 views

Max/min values attained by a function along a path

Say we have some arbitrary function $f(x,y) = xy$ or whatever. It doesn't have to be a scalar-valued function (though for my question it might be a restriction, not sure. Which is why I'm asking). If ...
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1answer
24 views

$1 \times 3$ tangent plane vs. gradient

Say we have some function $f(x,y,z)=xyz$ or whatever. It goes from $\mathbb{R}^3$ to $\mathbb{R}$. If we want the tangent plane of the function, we need to differentiate it, and this gives us a $1 ...
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1answer
27 views

Limits of $(e^{xy}-1)/y$

How would you solve this limit? It exists and is equal to 0 but I have no idea how to show it. $$\lim_{(x,y)\rightarrow(0,0)}\frac{e^{xy}-1}{y}$$
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529 views

Show that $f$ is constant on each sphere in $\mathbb{R}^3$ centered at the origin

Hi everyone this is a past exam question that I am studying as I go through my class that I am having trouble with, the full question is this: Let $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ be a ...
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2answers
68 views

How to plot a surface in maple where the range is given by an expression, not constants?

Im trying to plot the surface $z=(1+x^2)/(1+y^2)$ , but specifically the part of the surface that is above $|x|+|y|\leq1$. Cant seem to find any information on how to produce a plot in maple, where ...
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1answer
47 views

$f:R^2\to R$: Determining the Nature of a Critical Point when the Second Derivative Test Fails

I'm reviewing for a final exam tomorrow. This is an exercise that I am having trouble with: The function: $f(x,y)=x^2-y^4$ I determined that there is one critical point, at $(0,0)$. I determined ...
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94 views

Use Polar Coordinates to Find the Limit…

Use polar coordinates to find the limit. [If $(r, \theta)$ are polar coordinates of the point $(x, y)$ with $r \geq 0$, $r \to 0^+$ as $(x,y) \to (0,0)$)] $$\lim \limits_{(x,y) \to (0,0)} ...
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2answers
54 views

Extrema of two variable function

Find extrema of $f(x,y)=x^2-xy+y^2$ from set $M=\{ [x,y] \in \mathbb{R}^2;|x|+|y|\le1\}$ I am solving this kind of problems for the first time and I am not sure what I am doing, what I have got ...
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2answers
71 views

Multivariable calculus - find total derivative

I want to find the total derivative of the function $f: \mathbb R^n \to \mathbb R^n$, $f(x)=\frac{x}{|x|}$ If I was to copy what the teacher taught, I should find the limit of $\lim_{t \to 0} ...
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272 views

Surface integral on unit sphere

I'm struggling to calculate the surface integral in this question Find the area of the portion of the sphere $$z=\sqrt{1-x^2-y^2}$$ Which lies between the planes $z=0$ and $z=1$ Now I know the ...
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zeros and poles of meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$

I'm a bit confused regarding the following example: it is stated that $$ \frac{x^3+y^3+z^3}{x^2yz}$$ is a meromorphic section of $\mathcal{O}_{\mathbb{P}^2}(-1)$, and then one is asked to find poles ...
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48 views

How to compute the Jacobian Matrix of the next system?

I have a little problem with notation and I do not know how to work it out. I have the next system $$ \dot x = A(x)x, $$ where $x \in \mathbb{R}^n$ and the square matrix $A(x)_{n\times n}$ has ...
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1answer
34 views

Area of a triangle using double integration

What is the AREA of the triangle bounded by the lines $y=2x$ and $x=1$? I solved it using single integration and i got $A=1$. The problem is , I do not know how to solve it using double ...
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1answer
71 views

Trouble reading directional derivative proof

I'm reading Vector Calculus from http://mecmath.net/. This is a free PDF book for students of Calculus III. In section 2.4 (page 78) it introduces the directional derivative and theorem 2.2: So far ...
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155 views

a function with differentiable partial derivatives but unequal mixed derivatives

I am looking for an example of a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are both differentiable at some point, say ...
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31 views

What is $Df$ in Multivariable Calculus?

I wasn't in class for a week of lecture because of medical problems. It seems to do with differentiation but I cannot find how to find $Df$ given $f$ (don't even know what Df stands for) Here is the ...
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1answer
135 views

Hadamard variational formula Evans chapter 6 problem 15

This is Evans' chapter 6 problem 15. Consider a family of smooth, bounded domains $U(\tau) \subset \mathbb{R}^{n}$ that depend smoothly upon the parameter $\tau \in \mathbb{R}$. As $\tau$ changes, ...
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611 views

Find a possible equation for the linear function g(x,y) shown in the graph

Can someone please help me understand how to start this problem? I have posted this up before but have not received any help. I can obviously see that the gradient is 4, that the line passes ...
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1answer
79 views

Partial derivative operator expansion

Expand $$ (h\frac{\partial}{\partial t}+hf\frac{\partial}{\partial x})^3 $$ Where $$ x'(t)=f(x,t)=f $$ If this question is ridiculously hard to answer, at least tell me if this is correct $$ ...
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583 views

what to do when the multivariable second derivative test is inconclusive?

What do we do when the second derivative test fails? How do we approach it, and is there a general method to further find whether a critical point is a maximum, minimum or a saddle point? For ...
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74 views

Multivariable calculus - explain what the teacher did

The teacher gave this exercise: Find $D_f(a)$ when $f: \mathbb R^n \to \mathbb R$, $f(x)=<x,\xi>^2$ where $\xi \in \mathbb R^n$. What I did: I wrote it as $$f(x)= (\sum_{i=1}^{n}x_i ...
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49 views

How to resolve this diferential equation $y^2 y^{\prime}=x^3$

I see it is non-linear, but not sure if that is important here. I got the solution for the homogeneous in this way: $$y^2 y^\prime=0 \rightarrow y^\prime=0 \rightarrow \frac{dy}{dx}=0\rightarrow ...
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1answer
48 views

Surface integrals help?

I'm having trouble understanding visually how a surface integral works/calculates. For a standard double integral, function $f(x,y)$ and a rectangular region $U=[a,b]\times[c,d]$ then the double ...
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35 views

Can you use the chain rule when only one partial derivative is continuous?

Say the partial derivative with respect to $x$ of some function $f(x,y,z)$ exists and/or continuous, but the other partial derivatives don't exist. Can I still apply the chain rule if I'm only ...
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91 views

Taylor series expansion of a function with one independent and one dependent variable.

If $y=y(x)$ What will be the Taylor series expansion of $f(x+h, y(x+h))$ about $(x,y(x))$ upto first order partial derivatives of $f$. I think it will be , please help. ...
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1answer
34 views

Finding $\frac{\partial ^8 f}{\partial x^4\partial y^4}$

Given the function $f(x,y)=\frac{1}{1-xy}$ find the value of$\frac{\partial ^8 f}{\partial x^4\partial y^4}(0,0)$. First I developed the function into a taylor series using geometric series ...
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1answer
58 views

Evaluating a surface integral of a paraboloid

Calculate the average value of $(1+4z)^{3}$ on the surface of the paraboloid $z=x^{2}+y^{2}$,$x^{2}+y^{2} \leq 1$ I'm not sure on how to start this problem. I have already found the area of the ...
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2answers
322 views

volume inside sphere but outside hyperboloid

I am trying to find the volume inside the sphere $x^2 + y^2 + z^2 = 9$, but outside the hyperboloid $x^2 + y^2 - z^2 = 1$. by using a triple integral. for some reason i just cant seem to come up the ...