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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3
votes
1answer
144 views

Use chain rule to express $f_x$ and $f_y$ in terms of $f_\xi$ and $f_\eta$?

Consider partial differential equation: $f_x + 2 f_y = 1$ Trick is to introduce new variables $\xi = x$ and $\eta = 2x - y$ Using chain rule to express $f_x$ and $f_y$ in terms of $f_\xi$ and ...
3
votes
2answers
1k views

Find volume inside the cone $ z= 2a-\sqrt{x^2+y^2} $ and inside the cylinder $x^2+y^2=2ay$

I have this question and I have seen here that there are similar questions but I have been trying to figure it out for a while with no luck. I have to find volume inside the cone $z= ...
5
votes
1answer
74 views

Every closed $C^1$ curve in $\mathbb R^3 \setminus \{ 0 \}$ is the boundary of some $C^1$ 2-surface $\Sigma \subset \mathbb R^3 \setminus \{ 0 \}$

How can I prove it? This problem looks similar to Plateau's problem - but it is much more specific. I believe there exists some elementary proof. (Proving this will help me apply Stokes' theorem to ...
0
votes
3answers
103 views

Tangent plane of $z = e^{x\cdot y}\cdot \sin\left ( x^{2} + y^{2} \right )$

Assuming $$z = e^{x\cdot y}\cdot \sin\left ( x^{2} + y^{2} \right )$$ How can I find tangent plane equation @ $\displaystyle \left ( \frac{\pi }{2} , \frac{1}{\sqrt{2}}\right )$ ?
1
vote
1answer
138 views

Minimization of a cyclic sum with a certain function

Let $f:[0,2]^2 \rightarrow [0,1]$ be defined as $$f(x,y):=-\dfrac{x}{4} \sqrt{4-x^2}-\dfrac {1}{4 \sqrt 2}\sqrt{2(x^2+y^2)-x^2y^2+xy \sqrt{(4-x^2)(4-y^2)}}+\sin \dfrac ...
2
votes
1answer
124 views

Proving that the value of the integral doesn't depend on the surface

I'm trying to prove that if we have the vector field $v : \mathbb{R}^n \to \mathbb{R}^n$ given in spherical coordinates by: $$v(\rho, \theta, \phi)=\frac{1}{\rho^2}\hat{\rho}$$ Where $\hat{\rho}$ is ...
0
votes
1answer
281 views

Green's Theorem

Let $C$ be the closed,piecewise curve figured by traveling in straight lines between the points $(-2,1),(-2,-3),(1,-1),(1,5)$ and back to $(-2,1)$, in that order. Use Green's Theorem to evaluate the ...
7
votes
3answers
935 views

Surface integral over ellipsoid

I've problem with this surface integral: $$ \iint\limits_S {\sqrt{ \left(\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}\right)}}{dS} $$, where $$ S = \{(x,y,z)\in\mathbb{R}^3: \frac{x^2}{a^2} + ...
11
votes
3answers
611 views

Can the curl operator be generalized to non-3D?

In three dimensions, the curl operator $\newcommand{curl}{\operatorname{curl}}\curl = \vec\nabla\times$ fulfils the equations $$\curl^2 = ...
-2
votes
1answer
903 views

Evaluate the triple integral of $f(x,y,z)=\sin(x^2+y^2)$

Evaluate the triple integral of $$f(x,y,z)=\sin(x^2+y^2)$$ over the the solid cylinder with height $4$ and with base of radius $1$ centered on the $z$-axis at $z=−3$.
-1
votes
1answer
392 views

Evaluate the triple integral $\iiint_E \sqrt{x^2+y^2}dV$

Use cylindrical coordinates to evaluate the triple integral $$\iiint_E \sqrt{x^2+y^2}dV, $$ where $E$ is the solid bounded by the circular paraboloid $z=16−4(x^2+y^2)$ and the $xy$-plane. Please ...
-3
votes
2answers
95 views

Please Solve this Calc 3 Question.

Region $R=\left\{(x,y)|x≥0,y≥0,x+y≤1\right\}$ and $f(x,y)= x^2+y^3(x)$. A. Compute double integral $f(x,y)dA$ as iterated integral with respect to $y$ then $x$. B. Compute double integral $f(x,y)dA$ ...
0
votes
3answers
74 views

Evaluate $\int\limits_{-1}^1\int\limits_{-\sqrt{1-y^2}}^0\frac{1}{(1+x^2+y^2)^2}dxdy$

convert the double integral $$\int\limits_{-1}^1\int\limits_{-\sqrt{1-y^2}}^0\frac{1}{(1+x^2+y^2)^2}dxdy$$ to polar coordinates and then evaluate.
0
votes
1answer
59 views

Evaluate the Triple Integral

Use cylindrical coordinates to evaluate the triple integral $\iiint_E (x^2+y^2)dV$, where $E$ is the solid bounded by the circular paraboloid $z=16−4(x^2+y^2)$ and the $xy$-plane. This is Homework. I ...
4
votes
1answer
1k views

Finding the volume of a tetrahedron by given vertices.

Please help me with the problem below. Find the volume of a tetrahedron with vertices: $O(0,0,0)$, $A(1,2,3)$, $B(-2,1,5)$, $C(3,7,1)$ by using triple integral. Hint: First find the the equations of ...
1
vote
2answers
181 views

Finding the volume of $z = 81-x^2-3y^2$

Find the volume of the solid below the graph of the function $z = 81-x^2-3y^2$ above the region D in the xy-plane where D is the region between the parabola $y^2 = 2x+4$ and the line $y = x-1$.
1
vote
1answer
87 views

Polar coordinate

Let $f(x,y)$ be a differntiable function in $\mathbb{R}^2$ so that $f_x(x,y)y=f_y(x,y)x$ for all $(x,y)\in\mathbb{R}^2$. Find $g(r)$ so that $g(\sqrt{x^2+y^2})=f(x,y)$ and $g$ is differentiable in ...
1
vote
0answers
29 views

Prove limit without using operator norm

prove $$\lim_{h \to 0}\frac{f(\mathbf{x}+h)-f(\mathbf{x})-\langle\nabla f(\mathbf{x})h,h\rangle-1/2\langle\nabla^{2}f(\mathbf{x})h,h\rangle}{\|h\|^{2}}=0$$ where function $f:\Bbb{R}^{n} \to \Bbb{R}$ ...
0
votes
1answer
91 views

Lagrange multipliers for x,y,z

I have this question, I have run completely blind into. Find by Lagrange multipliers the volume V=xyz of where the largest box with sides adding up to x+y+z = k. I have found the gradient of V: ...
1
vote
3answers
825 views

Angle of Intersection for two curves?

Find the angle of intersection of ($r = 4 + \cos(3 \theta)$) and ($r = 4 -\cos(3\theta)$) at any point of intersection Can someone provide me a hint to express the curve as a parameterized curve in ...
1
vote
3answers
1k views

Gradient and Jacobian row and column conventions

Say $f$ is a scalar valued function from $\mathbb{R}^n \to \mathbb{R}$. When I learnt about the gradient $\nabla f(\mathbf{x})$ I always thought of it as a column vector in the same space as ...
1
vote
3answers
469 views

Solving a differential equation with more than one dependent variable

It's been awhile since I took differential equations. Now I am using differential equations in another class. This is why you shouldn't sell back books from your major courses. :) How would I solve ...
0
votes
1answer
239 views

Mean Value Theorem in $\mathbb R^n$ for discontinuity

For multivariable differentiation, it does not require continuity when when taking partial derivatives. Similarly, when they apply the mean value theorem in $\mathbb R^n$, one can have a function ...
-3
votes
1answer
827 views

Find the Average Value please

Find the average value of the function $f(x,y,z)=x^2+y^2+z^2$ over the rectangular prism $0≤x≤2, 0≤y≤2, 0≤z≤1$
2
votes
1answer
649 views

How to find limits of integration on a convolution of CRVs

In finding the convolution of two independent and continuous random variables, I am struggling with limits of integration. I cannot seem to figure out over what intervals the probability density ...
-2
votes
1answer
404 views

Volume of a pyramid.

Find the volume of the pyramid with base in the plane $z=−9$ and sides formed by the three planes $y=0$ and $y−x=3$ and $2x+y+z=3$.
1
vote
2answers
85 views

Evaluate $\iiint_E xyz \; \mathrm dV $

Evaluate the triple integral: $$\iiint_E xyz \; \mathrm dV$$ where $E$ is the solid: $0\le z\le 2, 0\le y\le z, 0\le x\le y$. Please explain this, I'm so confused.
0
votes
1answer
29 views

For harmonic function $U$, prove that $\int_{C_r}\partial_xUdy-\partial_yUdx$ is constant.

I need to prove that for a function $U$ harmonic in the punctured disk $0<r<r_0$, $\int_{C_r}\partial_xUdy-\partial_yUdx$, where $C_r$ is a circle of radius $r$, is a constant independent of ...
0
votes
1answer
51 views

Finding differentials of functions $p,q,v$

I am given the following 2 equations, where $p$ and $q$ are implicit functions of $v$. $$p^2 + vpq+q^2-1=0\\p^2+q^2-v^2+3=0$$ I need to find the values of $\large{\frac{dp}{dv}}$ and ...
2
votes
4answers
275 views

Evaluating $ \int_0^1 \int_y^1 \sqrt{1+x^2} dx dy $

$$ \int_0^1 \int_y^1 \sqrt{1+x^2} dx dy $$ I've tried switching the order as per fubini theorem to $\int_y^1 \int_0^1 \sqrt{(1+x^2)} dy dx$ and managed to get $\int_y^1 \sqrt{(1+x^2)} dx$ but am ...
0
votes
1answer
176 views

Divergence Theorem to compute volume bounded by paraboloid

Evaluate the integral $$\int_V \nabla \cdot \underline{r}\,dV$$ where $V$ is bounded by the surface $S_c$($S_c$ = part of the surface $z = a^2 - x^2 - y^2$ for which $z \geq 0$) and the plane $z=0$. ...
1
vote
1answer
205 views

Contour integration, Residue Theorem

Evaluate the integral using contour integration $$ \int_{0}^{\infty }\frac{dx}{1+x^a} $$ I know how to solve the integral $$ \int_{0}^{\infty }\frac{dx}{1+x^{2a}} $$ Using the residue theorem, ...
1
vote
0answers
113 views

Parametrizing a section of a torus

Consider the torus obtained by rotating the circle $(x-R)^2+z^2=r^2$ about the $z$-axis, where $R>r>0$. Parametrize the part of this torus where $z>x+y$. My approach to this so far is to ...
1
vote
1answer
442 views

Expected value of a multivariate distribution

Given this random vector: $$ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $$ And this probability distribution function that takes it as argument: $$ f_\mathbf{X}(\mathbf{x}) = ...
2
votes
1answer
62 views

Divergence Theorem to determine the flux

Please how can I determine the flux of the vector field $$F=(x+x^2 y){\bf e_x}+ (x y^2-y) {\bf e_y}$$ through the boundary which is formed by the the hyperbola $x^2 -81y^2=9$ and the lines $y=-2$ and ...
3
votes
2answers
483 views

Evaluate the Integral using Contour Integration (Theorem of Residues)

$$ J(a,b)=\int_{0}^{\infty }\frac{\sin(b x)}{\sinh(a x)} dx $$ This integral is difficult because contour integrals normally cannot be solved with a sin(x) term in the numerator because of ...
0
votes
1answer
69 views

Finding $u$ and $v$ in Jacobian substitutions

I've used Jacobians before in multivariable calculus to simplify integrals, but I'm lost when I need to find the substitutions myself. Today on the quiz, there was the problem $\int\int_{R} xy dxdy$ ...
1
vote
1answer
116 views

Finding partial derivatives of a 3 variable function

I am given the following function: $q(k,l,m) = k\,p(k,l) + m^2$ where $(k,l,m) \in \mathbb{R}$; $p(k,l)$ is a differentible function of $k$ and $l$;$k$ and $l$ are differentiable functions of $m$ ...
0
votes
1answer
80 views

Finding 1st order & 2nd order partial derivatives

I am given $v = p(a)$ and $a = q(t,u)$. How do I find $\frac{\partial v}{\partial t}$ and $\frac{\partial ^2v}{\partial t^2}$? I am thinking of the following: $$ \begin{align*} \frac{\partial ...
1
vote
1answer
146 views

$e^{F(x,y)}$ Type Multi-variable Exponential Integrals

I am sure all you integration buffs can do this faster than I can type it. Your help with a quick explanation and solution is appreciated. $$F _{XY} = \int_0^\infty\int_0^\infty xye^{-\frac{x^2 + ...
1
vote
1answer
355 views

What is the intuition behind the Lagrange multiplier?

I know that the minimum or maximum point is achieved when the gradient in the constraint function is parallel to the gradient on the $f$ function. But why the Lambda is called the Lagrange ...
0
votes
1answer
34 views

rate of change question I need some explanation

I tried to solve them myself but couldn't; the questions look kind of different from the explanation first given. Any help on how to approach this problems would be much appreciated. thank you a lot! ...
1
vote
0answers
90 views

Change of dependent variables

I am hoping to evaluate the integral $$ \int_0 ^R u(r,t)r^2\,dr$$ However, I do not have an expression for $u$, but rather $\tau(r,t)$, where the two are related by $$\frac{\partial\tau}{\partial ...
0
votes
1answer
54 views

What is the surface integral of the sureface

What is the surface integral of the sureface $S$ where $S$ is on the cylindrical surface $x^2+z^2=2az$ that cut by another surface $z=\sqrt{x^2+y^2}$. I couldn't even draw this thing out
1
vote
3answers
55 views

Answer Check: Show $L_1 = L_2$

Guys I need an answer check: Let $L_1$ and $L_2$ be two linear mappings from $R \rightarrow R^n$ safisfiying: $\lim_{h\to0} \frac{f(a+h)-f(a)-L(h)}{h} = 0$ prove that $L_1 = L_2$. Ok this is what ...
1
vote
2answers
71 views

Show $X-Y $ is open and $ Y-X$ is closed in $R^n$

$X$ is open in $R^n$ and $Y$ is closed in $R^n$ I understand that for Y to be closed $R^n - Y$ must be open, but I guess I'm confused on how to prove what an open set. I also understand that if $f$ ...
27
votes
5answers
1k views

The Meaning of the Fundamental Theorem of Calculus

I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
2
votes
1answer
84 views

What does this limit represent for this function?

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$, suppose we have the limit $$\lim_{(x,y)\to(a,b)} \dfrac{|f(x,y) - f(a,b)|}{\sqrt{(x-a)^2+(y-b)^2}}$$ The absolute value bars on the top is optional, but I ...
1
vote
1answer
82 views

Product rule for inner products using the 3 conditions

I understand there are multiple ways of of proving the product rule for the derivative of an inner product, though I cannot figure out how to do this one specifically: let $\alpha,\beta :R ...
4
votes
2answers
105 views

Show $\langle x,\nabla f \rangle = pf(x)$

I'm trying to figure out this problem. Perhaps Someone could give me some hints/solve it for me? It would be much appreciated. Let $U$ be an open subset of $R^n$ and suppose $f:U\rightarrow R$. Then ...