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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2
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1answer
22 views

How can I find these partial derivatives?

I'm reading a book which gives this function $f(x,y)=x^2y/(x^2+y^2)$ if $(x,y)\neq (0,0)$ and $f(0,0)=0$ as a $C^1$ function in $\mathbb R^2-\{(0,0)\}$, continuous in $(0,0)$ and it has the partial ...
-1
votes
1answer
33 views

find the area in the first quadran

By first making an appropriate transformation, find the area in the first quadrant which is bounded by the curves $x = y, x = 2y, xy = 1$ and $xy = 2$ answer $u=x \times y$; $v=\frac{y}{x}$ (is ...
1
vote
0answers
33 views

First fundamental theorem of calculus for line integrals [on hold]

Please, could someone look at this tricky question? Find the work done by force $F(x,y)=(3y^2+2) \hat i+16x \hat j$ in moving a particle from $(-1, 0)$ to $(1,0)$ along the upper half of the ellipse ...
0
votes
1answer
16 views

Prove that Euler's equation can be written in a specific form

According to my notes, the following theorem holds: If $y$ is a local extremum for the functional $J(y)= \int_a^b L(x,y,y') dx$ with $y \in C^2([a,b]), \ y(a)=y_0, \ y(b)=y_1$ then the extremum $y$ ...
0
votes
0answers
9 views

Taylor series for a multivariable function

We know the following: $a \approx 1 + x\sqrt{dt}$ $V_1 = V(aS, t + dt ) $ The textbook claims you can (using Taylor's Theorem), expand the bottom-most equation like this: $V_1 \approx V + ...
0
votes
1answer
38 views

Sketch this body and calculate the mass of the body

A solid body in the first octant is bounded by the planes $z = 0$, $y = 0$, $z = y$ and the parabolic cylinder $y = 4 − x^2$, and has density $\rho = xz$. calculate the mass of the body. answer ...
2
votes
1answer
39 views

Evaluating a triple integral by inspection

I would like to evaluate the triple integral: $$\iiint\limits_D {2 + 3{x^2} + 3{y^2}dV}$$ where $D$ is a conic domain with vertex $(0,0,b)$ and axis along the $z$-axis with a base (disk) with radius ...
2
votes
1answer
21 views

Counterexample to the double integral computational theorem when the double integral existence assumption is dropped?

To make things simple, consider the simplest case of the double integral computational theorem. Throughout any phrase involving integrability is in the Riemannian sense. Let $[a, b], [c, d] \subset ...
1
vote
1answer
19 views

Volume inside region delimited by surfaces $z=5-x^2$, $z=y$ and $y=1$.

I need to find the volume inside the region $E$ delimited by surfaces $z=5-x^2$, $z=y$ and $y=1$. I've spent few hours on this and would really need a hint from a charitable soul. I see that the ...
0
votes
2answers
40 views

I need help with the integration order please

the integral is as follows: find the volume between these regions bounded by : $z = x^2 + 3y^2$ and $z = 9 - x^2$ I discovered that this would be the space bounded by the elliptic paraboloid and the ...
0
votes
0answers
7 views

Certain local inequality for volume and surface measures

Suppose $S$ is closed simple piecewise smooth curve for in the plane (It is viewed as boundary of a domain). Does the following hold ...
0
votes
0answers
26 views

Calculating the magnetic vector potential

I've calculated A as proportional to r^2 for ra but I really don't think is correct. If someone could take me through the calculation I would really appreciate it.
0
votes
0answers
17 views

Using polar coordinates in this integral

I'm trying to solve something along the lines of: $$\iint \frac{\partial F_1(x,y)}{\partial x}+\frac{\partial F_2(x,y)}{\partial y}dydx$$ which I want to change to polar coordinates, but I don't ...
1
vote
1answer
34 views

Critical Point of $\mathbb{R}^2 \to \mathbb{R}^2$ function

Given a function $f:\mathbb{R}^2 \to \mathbb{R}$ I can find critical points by finding the $1\times 2$ Jacobian matrix, setting each partial derivative equal to zero and solving the equations. I can ...
0
votes
1answer
18 views

Trying to integrate the volume of a body

I was trying to integrate the volume of a body blocked by $z=0$, $z=2x$, $x+y=3$ and $y=0$ using the double integral... but I don't really know how to approach this.
1
vote
0answers
34 views

Integration of a symmetric function

I have a bit of confusion about the following situation. Let's assume that we have a symmetric function $f(x,y)$ where it has the property $f(x,y) = f(y,x)$ for all $x$ and $y$. $x$ and $y$ have the ...
0
votes
2answers
38 views

Volume by double or triple integral?

I was trying to integrate the volume of a body blocked by $z=0,\; z=2x,\; x+y = 3$ and $y=0$ using the double integral... however it didn't work yet. I'm convinced its a double integral and not a ...
0
votes
0answers
14 views

Proving that average value of $u$ around a circle is the value of $u$ at the centre.

I would like to prove that: If $u(x,y)$ is harmonic in a domain containing a disk of radius $r$ with boundary $C_r$ $\implies$ the average value of $u$ around the circle is the value of $u$ at the ...
2
votes
1answer
36 views

Trying to integrate $\iint_D x^4\tan(x)+3y^2 \,dA$.

I'm trying to integrate $\iint_D x^4\tan(x)+3y^2\, dA$ in domain $D=\{(x, y) \in \Bbb R^2 \mid x^2+y^2\le4, y\ge0\}$. Domain is simple enough; half circle of radius 2 over $x$ axis. Converting to ...
0
votes
1answer
27 views

Partial derivatives after a change of variables

Say I have a function of $n$ variables $F(x_{1}, x_{2}, x_{3},...,x_{n})$, where $x_{1} = g_{1}(y_{1}, y_{2}, y_{3},...,y_{m})$, $x_{2} = g_{2}(y_{1}, y_{2}, y_{3},...,y_{m}),\dots, x_{n} = ...
0
votes
0answers
21 views

Maximizing the following function

I need to find values of $k_1$, $k_2$ and $k_3$ that maximize $C^{m_1}_{mm_1} \cdot C^{m_2}_{k_1-mm_1} \cdot C^{n_1}_{nn_1} \cdot C^{n_2}_{k_2-nn_1} \cdot C^{p_1}_{pp_1} \cdot C^{p_2}_{k_3-pp_1}$ ...
2
votes
0answers
56 views

Understanding a step in Yi Fang's Lectures on Minimal Surfaces

In Yi Fang's Lectures on Minimal Surfaces, page $94$, there's a step that I didn't understand, and that perhaps is wrong. I'll estabilish some notation first. We have that $X$ is a minimal surface, ...
1
vote
3answers
48 views

Prove that $f(x,y)$ is continuous in $(0,0)$

Prove that $f(x,y)$ is continuous in $(0,0)$, where \begin{equation} f(x,y) = \begin{cases} \frac{x^2y}{x^4+y^2}, & (x,y)\neq 0\\ 0, & (x,y) = (0,0) \end{cases} \end{equation} The solution I ...
0
votes
0answers
11 views

formula to establish correlation between multiple library functions

I am trying to predict the change in timeliness of holds delivery relative to number of owned Bestsellers, number of holds and the checkout window. (yes, it really is a library question). To do this ...
0
votes
0answers
27 views

How to prove the relation $\tan \theta = \hat{\vec{n}} \cdot \nabla h$

The relation for finding the contact angle is often given as $\tan \theta = - \hat{\vec{n}} \cdot \nabla h$ in papers such as in Sequential deposition of overlapping droplets to form a liquid line ...
1
vote
0answers
24 views

Find the mass flow rate, given a surface, density and velocity field

I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post). They ask me to find the mass flow rate passing through a surface, where the velocity field is ...
1
vote
0answers
10 views

Finite cover of balls for a bounded subset

Suppose X is a bounded subset of a k-dimensional manifold $M$ $\subset$ $R^n$ with $(k-1)$-dimensional volume $0$. I need to show that this implies for all $d>0$, all points of X can be contained ...
1
vote
1answer
51 views

Vector calculus identity for $\nabla\times(\vec{b}\cdot\nabla)\vec{b}$

I'm going through a paper on turbulence and in it the author uses the following $$ ...
1
vote
2answers
31 views

Convergence of a double integral

Is the integral $$\int_1^\infty\int_{e^{-x}}^1\frac{\sin y}{x^2y}dy dx$$ convergent or divergent?
0
votes
0answers
38 views

surface integral of the normal component of $curl(\mathbf{F})$

I have a square with side lengths $a$ in the $xy$-plane, and we build he square up vertically in the $z$-direction to construct a cube. What is the surface integral of the normal component of the ...
1
vote
1answer
24 views

Finding domain, range, and level curves of $f(x,y)=\arcsin(6y-5x)$

I will like it if someone helps me out and also checks my work for me for this equation: $f(x,y)=\arcsin(6y-5x)$ Domain $-1\leq6y-5x\leq1$ $y\geq\frac{5x}{6}-\frac{1}{6}$ and ...
2
votes
1answer
35 views

$f(x,y)=4x^3y^2$ Directional Derivative…

Let $f(x,y)=4x^3y^2$ How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most ...
2
votes
3answers
65 views

Finding Extrema of $f(x,y)=x^4+y^4-4xy$

Let $f(x,y)=x^4+y^4-4xy$ How do I find all the relative extrema and saddle points of $f$ which lie within the open square ${(x,y) | -2<x<2,-2<y<2}$. And also if $f$ was in the closed ...
1
vote
0answers
17 views

$f(x,y)=4x^3y^2$ Dealing with Directional Derivatives and Vectors

Let $f(x,y)=4x^3y^2$ How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most ...
0
votes
1answer
29 views

Proving that $f(x)=0\ \forall x\in B(0,r)$

Let $y=f(x_1,…,x_n)$ be differentiable on $B(0,r)$. Assume that $\dfrac{\partial}{\partial x_i}f(x)=0\ \forall x\in B(0,r)$ and $i\in\{1,…,n\}$. How to prove that $f(x)=0\ \forall x\in B(0,r)$? Do ...
-1
votes
0answers
23 views

Computing $f_{xy}$ and $f_{yx}$ [duplicate]

Let's consider the following function: $$f(x,y)=\begin{cases} xy\left(\dfrac{x^2-y^2}{x^2+y^2}\right) & \text{if }x^2+y^2\not=0 \\{}\\ 0 & \text{if }x=y=0 \end{cases}$$ I need to compute ...
0
votes
2answers
27 views

$f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$

I think that I understand what the question wants me to do: $f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$ I worked out the partial derivatives: ...
1
vote
2answers
23 views

Plot the level curve of sine function in multiple variables

I'm very confused about how I could go about this, as it seems that the question cannot be done using only the information given. The question is: plot the level curve for $f(x,y) = \sin(k^2x^2 + ...
1
vote
1answer
56 views

Proving that $f$ is differentiable at $0$

Let's consider the following function: $$f(x,y)=\begin{cases} (x^2+y^2)\sin\left(\dfrac{1}{x^2+y^2}\right) & \text{if }x^2+y^2\not=0 \\{}\\ 0 & \text{if }x=y=0 \end{cases}$$ I know that ...
1
vote
2answers
41 views

Find a unit vector that is parallel to $\nabla f(\cos\theta,\sin\theta)$

Suppose $f(x,y)$ is differentiable for all $(x,y),f(x,y)=17$ on the unit circle $x^2+y^2=1$, and $\nabla f$ is never zero on the unit circle. For any real number $\theta$, I have to find a unit vector ...
1
vote
0answers
18 views

Volume of the solid between these two parabaloids - can someone verify my answer?

I wish to find the volume of the solid bounded by the surfaces: $z = x^2 + y^2$ and $z = 4 - a^2x^2 - a^2y^2$. I set the two surfaces equal and it gave me a circle, so, I used cylindrical ...
0
votes
0answers
35 views

Find Directional Derivative, Unit Vector, and Rate of Change

Let $f(x,y)=4x^3y^2$ How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most ...
3
votes
1answer
25 views

Lagrange multipliers method - absolute maximum and minimum

Using the Lagrange multipliers method I have to find the absolute maximum and minimum value of $f(x, y)=x^2+y^2-x-y+1$ in the unit disc. So, I have to find the extremas of $f(x, y)=x^2+y^2-x-y+1$ ...
2
votes
2answers
57 views

How to evaluate $\sum_{j=0}^\infty\;\sum_{\substack{k=0 \\ k \neq j}}^\infty \frac{1}{j^2-k^2}$

I was reading an introductory text on multiple integrals and I have encountered a problem asking me to explain why $$ \sum_{j=0}^\infty\;\sum_{\substack{k=0 \\ k \neq j}}^\infty ...
2
votes
1answer
18 views

Determine the volume of a solid given specific bounds

Determine the volume of the solid enclosed by the paraboloid $z = x^2 + y^2$ and the plane with equation $4x − 2y + z = 0$. Could someone explain to me whether I use double integral polar coordinates ...
0
votes
1answer
20 views

Parameterise the path C of a square

I have a question, I am required to parameterise the square with side lengths $a$, going in a counterclockwise direction. I have determined then that the points are $$(0,a), (a,0), (0,0), (a,a)$$ ...
6
votes
0answers
66 views

Very difficult surface integral

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}), \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
1
vote
2answers
50 views

Are partial derivatives a special case of the total derivative or just something else entirely?

I can do basic multivariable calculations using partial and total derivatives. I also know for partial derivatives the existence of all partial derivatives at a point doesn't imply continuity. Are ...
1
vote
0answers
5 views

Conservative field $F$ on not simple connected set

Give an example of a field $F:D\subseteq \mathbb R^2 \to \mathbb R^2$ such that $D$ is a doubly connected set (that is $D$ has on "hole") but $F$ is conservative And if $D$ is a triply connected ...
1
vote
0answers
34 views

Prove or Disprove that if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ both exist, then f is continuous at $(x_0,y_0)$ [duplicate]

I have to either prove or disprove the fact that if $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ both exist, then f is continuous at $(x_0,y_0)$. What I thought: I thought that the best way to approach this is ...