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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4
votes
1answer
48 views

Solving $\int_{0}^{1}\int_{0}^{1-y} \sin\frac{x-y}{x+y}\mathrm dx\mathrm dy$

How would I go about solving the following double integral? $\int_{0}^{1}\int_{0}^{1-y} \sin\frac{x-y}{x+y}\mathrm dx\mathrm dy$ I am absolutely clueless on what to do with that sine.
-1
votes
1answer
37 views

Finding min and max under constraints

I have a two variable function: $f(x,y)=4x^2-y^2-xy-2x+6y$. I need to find its absolute minimum and maximum under the constraints: $y=4-2x$, $x \geq 0$, and $y \geq-2$. I am not sure how to do it, ...
4
votes
3answers
35 views

Coupled differential equation arising in flow line.

So, I ran (certainly not literally) across these two coupled differential equation given by: $$x'(t)=\left(x(t)\right)^2-\left(y(t)\right)^2 $$ $$ y'(t)=2x(t)y(t)$$ These equation occurred ...
0
votes
0answers
27 views

Decreasing of power function.

Show that $\frac{3}{4}(x-2)x^{-\frac{5}{2}}-(x+1)^{-\frac{3}{2}}<0$, wherer $x\ge0$. I tried by taking differentiation but then expression become more complicated. I also tried by checking the ...
2
votes
0answers
23 views

Is my graph correct?

My problem told me that: Let $S$ be the surface described by the equation in cylindrical coordinates $z = r^2$, and the inequality $0 ≤ z ≤ 4$, oriented such that the unit normal vector points ...
1
vote
3answers
57 views

What is the minimum point of $x\mapsto x^Ty$ for $|x|\le 1$ and a fixed $y\in\mathbb{R}^n$?

Let $y\in\mathbb{R}^n$. I want to minimize $$f(x):=x^Ty\;\;\;\text{for }|x|\le 1$$ The minimum point should be $$-\frac{y}{\sqrt{y^Ty}}\tag{1}$$ However, how can we derive $(1)$ analytically? Since ...
3
votes
1answer
77 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Show that $\forall ...
3
votes
0answers
20 views

Any unit speed reparametrization $\beta=\alpha(h)$ of $\alpha$ is reparametrized by an arc length function.

These are definitions given from Barret O'neill's Elementary Differential Geometry. Definition $1$. A curve in $\mathbb{R}^3$ is a differentiable function $\alpha: I\to \mathbb{R}^3$ from an open ...
0
votes
1answer
38 views

water level in a tank

I am trying to develop a relationship that can give the area fraction of a circle that contains water, as a function of the water level, (h) Say I have a unit circle $1-x^2-y^2$ and set the bounds ...
1
vote
0answers
18 views

Drawing for multivariable calculus.

Does anyone have any step-by-step instructions for how they typically go about drawing 3d images, slices, and projections. I'm having a really hard time looking for good outlets/examples on how to ...
0
votes
1answer
81 views

Can anyone help me with this double anti-derivative? None of my teachers can…

The integral is the following: $$\iint\sqrt{r^2-x^2-z^2}dxdz$$ Where r is a constant and x and y are both variables. I have filled pages and pages trying to solve it, i think it can be solved by ...
1
vote
1answer
29 views

Finding Extreme Values (Multivariable)

Given $f(x,y)=x^2+2y^2$, find its extreme values on $x^2+y^2=1$. I know how to solve this problem using Lagrange's method and the constant variation method. The solutions are $(\pm1,0)$ and ...
0
votes
1answer
27 views

Multivariable calculus & partial derivatives problem

Let $$H = \mathbb f(S,V) $$ $$\dfrac{\partial H}{\partial S}S\sqrt{V} = \mathbb g(H) $$ $$\dfrac{\partial H}{\partial V}\sqrt{V} = \mathbb h(H) $$ Note that functions $\mathbb g$ and $\mathbb h$ ...
2
votes
3answers
67 views

Continuity of a function $f: \mathbb{R}^2 \to \mathbb{R}$

It's easy to check that the function $$ f_1(x, y) = \begin{cases}\frac{x y}{x^2 + y^2} &\text{if (x, y) ≠ (0, 0)}\\0&\text{if (x, y) = (0, 0)}\end{cases}$$ is not continuous in $0$, because ...
0
votes
0answers
19 views

Partial derivatives and chain rule explanation.

I have a function $w=f(x,y)$, where $x=r\cos{\theta}$ and $y=r\sin{\theta}$ and I'm asked to show that $$\frac{\partial w}{\partial x}=\frac{\partial ...
0
votes
2answers
37 views

What's the parametric equation for the plane through a point (x,y,z) perpendicular to (a,b,c)?

Find the parametric vector and Cartesian equations for the following planes: a. The plane thru point $(2,1,-2)$ perpendicular to vector $(-1,1,2)$. b. The plane thru the three points $(2,2,-2)$, ...
0
votes
0answers
17 views

Linear Algebra requirement Spivak's Calculus on Manifolds

I am interested in the extent of knowledge of Linear Algebra required for Spivak's Calculus on Manifolds. More precisely, in the first problems in his book they reference norm preservation and ...
1
vote
1answer
25 views

Determine if first and second partial derivatives are positive, negative or zero based on level curves

Assuming I have a point on a level curves graph for function f(x,y), how would I determine whether the first and second partial derivatives are positive, negative, or zero? I understand that for a ...
1
vote
0answers
76 views

Find the area portion of Surface of $x^2+y^2+z^2 =b^2$ inside a cylinder

Find the area of the surface portion: $x^2+y^2+z^2=b^2$ that remains within the right cylinder $ x^2+y^2=by$ , where $b > 0$
0
votes
0answers
50 views

Surface integral of $x^2+y^2+z^2 =b^2$ inside a cylinder

I need find the area of the surface portion: $x^2+y^2+z^2=b^2$ that remains within the right cylinder $ x^2+y^2=by$ , where $b > 0 $ I understand that I must find the whole surface that rises ...
0
votes
1answer
43 views

Is this a sufficient condition for differentiability

Consider a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$. Suppose that, for all $c\in \mathbb{R}$, every vector in $f^{-1}(c)$ is supported by a unique hyperplane to $f^{-1}(c)$. Is $f$ ...
2
votes
4answers
53 views

What is the physical meaning of this integral?

Let $$I=\int_S ~z~dS$$ Where $S$ is the surface of a hemisphere with equation $x^2+y^2+z^2=4~~~~z \geq0$. I know $$\int_S~dS$$ would be the surface area of the hemisphere but I can't figure out how ...
3
votes
2answers
39 views

Differential equations: IVP application

I am given $$y'=0.05y-800$$ I am asked to: (a) Find all constant solutions of the differential equation. (b) Suppose $y = M$ is your constant solution from (a). Plot two solutions of the ...
2
votes
1answer
26 views

Give a formula for the volume of the solid under a surface $z=xy$ and a triangle?

Given is the solid with unit density lying under the surface $z = xy$ and above the triangle in the $xy$-plane with vertices $(0, 1, 0)$, $(1, 1, 0)$ and $(0, 2, 0)$. Give a formula for the ...
5
votes
1answer
33 views

$x_t := a_t -b_t c_t $ , with $dx_t = \theta (\mu-x_t) dt+ \sigma dW_t$

I would like to solve the following equation explicitly using Ito's lemma: $$ x_t := a_t -b_t c_t , $$ where $x_t$ is an Ornstein-Uhlenbeck process (see here) $$ dx_t = \theta (\mu-x_t) dt+ \sigma ...
1
vote
0answers
15 views

Inverse of a Bijective Bivariate quadratic function or polynomial

I am looking for some general way to invert a bijective quadratic polynomial of the form $$ f(x,y)=A_0x+A_1x^2+Axy+B_0y+B_1y^2+Byx $$ where the coefficients may or may not be in the same ring as the ...
3
votes
3answers
30 views

Find function $f: U \subset \mathbb{R}^2 \to \mathbb{R}$ with $||Df(x,y)|| \leq 1$ such that $P,Q \in U$ exist with $|f(P) - f(Q)| > || P-Q||$

I am looking for a continuously-differentiable function $f: U \to \mathbb{R}, U \subset \mathbb{R}^2$ which satisfies the following requirements: $U$ is open set $U$ is connected (right word?), i.e. ...
1
vote
2answers
46 views

Proof that a log-of-sum-of-exponentials is a convex function

It's well known in statistical mechanics that the following is a convex function of the vector $\theta$: $$ A(\theta) = \log \left( \sum_{i=1}^\infty e^{\theta \cdot f(i)} \right) $$ where $f(i)$ is ...
1
vote
1answer
43 views

Lagrange multipliers problem with two constraints

Hi guys I am working with the following polynomial and I am trying to find the $\lambda , \mu$. I have a polynomial and I am trying to do Lagrange multipliers. Here is what I have. $f(x,y,z)= a ...
2
votes
2answers
32 views

Multivariable Calculus -Jacobian of the transformation

I am trying to figure out the answer to this problem: Evaluate $$\iint sin(\frac{x+y}{2}) cos(\frac{x-y}{2})dA$$ on $R$, where $R$ is the triangle with vertices $(0,0),(2,0)$ and $(1,1)$. I ...
0
votes
1answer
37 views

Double integrals: How to choose the order of the limits of integration

I have to calculate the integral of $|x|$ in the region that you see here: I have some doubts about the point from which start to integrate and about the order that I have to follow for the limits ...
-1
votes
1answer
71 views

Double integration of $\frac{1}{\sqrt{x^2 + y^4}}$

I am just learning double integration. I am stuck with the following problem: $$\int_{\mathbb{R}^2}\frac{1}{\sqrt{x^2 + y^4}}\,dx\,dy$$ I am not even sure whether is integral is finite. I would ...
-4
votes
2answers
56 views

numerical techniques of integration.

I am working on some past exam questions of integration and i came across this question. Can any body solve & explain this in detail to me. Thanks using FTC we Know $$\int_1^3 \frac{1}{x^2} = ...
-1
votes
0answers
16 views

How to sketch this region in $3$ dimensions?

I need to sketch the following regions but I'm having major difficulty in doing so. Could anyone please help me. I don't want the full answer but want to learn how I can do it thanks. $$R=\{(x,y,z):0 ...
1
vote
2answers
48 views

Differentiability of $f(x,y)=2xy+\frac{x}{y}$ at $(1,1)$

I'm trying to prove that the function $$f(x,y)=2xy+\frac{x}{y} $$ is differentiable at $(1,1)$. So I got: $$\nabla f(1,1)=\begin{bmatrix}3\\ 1 \end{bmatrix}$$ and: ...
1
vote
2answers
24 views

Computing partial derivatives using three implicitly defined equations

The three equations $x^2-y\operatorname{cos}(uv)+z^2=0$ $x^2+y^2-\operatorname{sin}(uv)+2z^2=0$ $xy-\operatorname{sin}u\operatorname{cosv}+z=0$ define $x,y,z$ as functions of $u,v$. Compute the ...
1
vote
2answers
31 views

Find a unit tangent vector to a curve that is an intersection of two surfaces.

The intersection of the two surfaces given by the Cartesian equations $2x^2+3y^2-z^2=25$ and $x^2+y^2=z^2$ contains a curve $C$ passing through the point $P=(\sqrt{7},3,4)$. These equations may be ...
0
votes
1answer
29 views

Showing that $\nabla (\alpha f) = \alpha \nabla f$ for constant $\alpha$

I want show that del of alpha times a vector function for is equal to alpha times del of fun using. Alphar is a constant hence it should be factories out after finding partial derivetives,but how do ...
0
votes
1answer
37 views

If the integral of a vector field over a closed curve equals zero, is the field conservative?

If a vector field has a potential, then the integral of that vector field over every closed curve is zero. If the integral of a vector field over a closed curve equals zero, does that imply that the ...
0
votes
1answer
25 views

Differential of a vector field

How is the derivative of a vector field defined? Gradient only works on scalar fields, divergence or rotation is not what I am looking for. Let's take an easy example: $f: \mathbb R^2 \to \mathbb ...
3
votes
2answers
35 views

relation between $\frac{\partial(x,0)}{\partial x}$ and $\left.\frac{\partial(x,t)}{\partial x}\right|_{t=0}$

if $u(x,t)$ differentiable function and i only have $u(x,0)$, then is it right $\frac{\partial(x,0)}{\partial x} = \left.\frac{\partial(x,t)}{\partial x}\right|_{t=0}$ or can i derive $u(x,0)$ to $x$ ...
0
votes
1answer
26 views

How to determine the maximum rate of increase in temperature

Suppose that the temperature at a point $(x,y,z)$ in space is given by $T(x,y,z)=\frac{80}{1+x^2+2y^2+3z^2}$ where $T$ is measured in degrees celsius and $x$,$y$ and $z$ in meters. In which ...
0
votes
0answers
13 views

Regular level set theorem

Let $F:N \rightarrow M$ be a $C^\infty$ map between two smooth manifolds both of finite dimensions and a chart $(V,\psi) = (V, x^1,...,x^m) $ centered in a point $p \in M$. Why $F^{-1}(V) $ contains ...
1
vote
1answer
17 views

Null velocity and piecewise smooth path

On texts of multivariable calculus and real analysis I have always seen the work made by $\mathbf{F}$ along the path $\gamma$ defined as the ...
0
votes
1answer
41 views

What is the geometrical interpretation of these three integrals?

What is the geometrical interpretation of these three integrals? 1) $$\int_C~ (x+y) ~dx + (x^2y)~ dy$$ 2) $$\int_C~ \vec{F}\cdot d \vec{r}$$ 3) $$\int_C~ xyz~ ds$$ I know they are all line ...
2
votes
1answer
34 views

Change of variables in a double integral-proving injectivity

I want to make the following change: $$u=x^2 - y^2 \quad v=xy$$ where my region is in the first quadrant bounded by: $x^2-y^2 = 3 , \quad xy=1 , \quad x^2-xy-y^2 = 1 $ . How can I prove this change ...
2
votes
1answer
25 views

Showing $∇f (0, 0) = (0, 0)$ using chain rule

I'm trying to show the following but I'm not very sure how to proceed. Could someone please explain to me how to approach and solve the following question? Let $f : \Bbb R^2 → \Bbb R$ be ...
2
votes
1answer
40 views

Flux integral using Cartesian coordinates

Problem - need help for part (ii) Let $\vec{F} = y \vec{i} -x \vec{j} + z \vec{k}$ and let the surface $S$ be the part of the paraboloid $ z = 4 - x^2 - y^2$ with $z \geq 0 $, oriented with $\vec{n}$ ...
0
votes
1answer
16 views

Vector-valued function to describe a hyberboloid

I need to find a vector-valued function to describe the quadric surface $x^2+y^2-z^2=1$. I could use the identity $\cosh^2 u - \sinh^2 u = 1$, but I'm not sure how. The best I could arrive at is ...
2
votes
1answer
74 views

Integration of $|y|^{-2}$ over the ball $B(0,r)$

Can one explain why taking an integral of $\frac 1{|y|}$ over a ball in $\mathbb{R}^3$ of radius $r$ is equal to a constant times $r^2$? If $y \in \mathbb{R}^3$, then $$\int_{B(0,r)} ...