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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17 views

Filling in Stewart's derivation of the arc length formula

To derive the formula for the arc length of a curve in 3 dimensions the approach Stewart takes is to estimate the length by summing the lengths of n subintervals. Then the actual length is assumed to ...
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1answer
12 views

Fundamental Period of a Summation

How would I find the fundamental period of a summation such as this one? Any hints/suggestions?
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1answer
27 views

Is $\frac{1}{xy}$ convex for $x,y>0$

Is the function $$f(x,y) = \frac{1}{xy}$$ convex for $x,y>0$. I computed the hessian but it is very complicated and I do not know how to show it is positive semi definite.
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0answers
20 views

Showing that $I(\xi_1,…,\xi_d)=0$

Let $\xi_1,...,\xi_d \in S^{d-1}$ and $Leb(B)=0$. We define $$I(\xi_1,...,\xi_d) = \int_0^\infty \cdots \int_0^\infty 1_B (r_1 \xi_1+\cdots+r_d \xi_d) \prod_{j=1}^d g(\xi_j,r_j) (r_j^2 \wedge 1) ...
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1answer
12 views

Surface integral over the surface of the cone $z=1-\sqrt{x^2+y^2}$ lying above the $xy$-plane and normal making an acute angle with $\vec k$

Let $\vec F=(x^2+y-4,3xy,2xz+z^2)$ and $S$ be the surface of the cone $z=1-\sqrt{x^2+y^2}$ lying above the $xy$-plane and $\vec n$ is the unit normal to $S$ making an acute angle with $\vec k$ , then ...
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1answer
13 views

certain durface integral on the surface of the solid bounded by the sphere $x^2+y^2+z^2=10$ and the paraboloid $x^2+y^2=z-2$

Let $S$ be the surface of the solid bounded by the sphere $x^2+y^2+z^2=10$ and the paraboloid $x^2+y^2=z-2$ , let $\vec F=(4xz,-y^2,4yz)$ , then how to evaluate $\iint_S\vec F.\vec n dS$ , where $\vec ...
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2answers
57 views

Prove that $f:\mathbb{R}^n\to\mathbb{R}$ is continuous

Let $f:\mathbb{R}^n\to\mathbb{R}$ such that for every continuous curve, $\gamma:[0,1]\to\mathbb{R}^n$: $f\circ\gamma$ is continuous. Prove that $f$ is continuous. So I know we shall prove it by a ...
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0answers
9 views

Secant equation

As I am reading the book (Nocedal & Wright: Numeric optimization), I've encountered an equation: $$B_{k+1} s_k = y_k$$ that is, "Hessian at the new point times the step equals to the function at ...
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1answer
32 views

Finding Surface Area of A Right Cone with Calculus

I'm really struggling with this question and I think a lot of it has to do with the visualization of it (I've drawn pictures but for some reason I'm still having trouble). Work: Let R be the radius ...
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1answer
49 views

Bounding the remainder

I need to find the 3rd order Taylor polynomials and bound the remainder term at $(0,0)$. The function is $$f(x,y)=\cos(x)\sin(y)$$ Here is what I did: first, I found the taylor expansions of sin and ...
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0answers
27 views

Is the following function continuously differentiable?

I am given a piecewise function, $f(x,y)=(xy,\frac{x^4}{x^2+y^2})$ if $(x,y) \neq 0$ and $f(x,y)=(0,0)$ if $(x,y)=0$. Thus $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am asked if this is ...
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1answer
56 views

Existence of a Function?

I'm trying to find a smooth function $f: \mathbb{R} \times (0, \infty) \rightarrow \mathbb{R}$ which satisfies the following conditions: 1) $\lim_{t \rightarrow 0^+} t\cdot \left(\frac{\partial ...
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1answer
55 views

Parametrize an intersection of a plane and an elliptic paraboloid

I'm supposed to parametrize the intersection of the plane that has the equation $z = 5x + 3y$ and the 'elliptic paraboloid' with the equation $z = 3x^2+2xy+3y^2$ These two equations can also be ...
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1answer
36 views

Can any smooth function be written in this form?

Can any smooth function $F: \mathbb{R}^n \to \mathbb{R}$ be written in the form$$F(x) = F(a) + \sum_{\mu = 1}^n (x^\mu - a^\mu)H_\mu(x),$$where $a = (a^1, \dots, a^n) \in \mathbb{R}^n$ and the $H_\mu$ ...
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1answer
21 views

How to solve the following multi-variable integral? [closed]

How can I solve it? click here to see the question
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0answers
14 views

Multivariable calculus, Great Circle. Intersection between a plane and a sphere

"A great circle on a sphere is a circle that consists of the intersection of the sphere with a plane through the center of the sphere. View the unit sphere S with equation $x^2+y^2+z^2=1$ and the ...
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1answer
28 views

How to evaluate $\iint_R \cos (\max \{x^3,y^{3/2}\}) dx dy$ , where $R:=[0,1]\times [0,1]$ ? [closed]

How to evaluate $\iint_R \cos (\max \{x^3,y^{3/2}\}) dx dy$ , where $R:=[0,1]\times [0,1]$ ? I tried breaking the region to do case by case , but I am not getting anywhere . Please help . Thanks in ...
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1answer
85 views
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1answer
44 views

Finding minimum of a two variable function

Let $D=\{(x,y)\in\Bbb R^2:1\le x\le1000,1\le y\le1000\}$. Define $$f(x,y)={xy\over2}+{500\over x}+{500\over y}$$ Then the minimum value of $f$ on $D$ is Finding $f_x=\frac y2-{1000\over ...
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0answers
72 views

How do we know that an integral is unsolvable?

I am currently learning intro differential equations. I am confused how one knows that an ODE will not be solvable. It seems that for the most part, the equations becomes "unsolvable" about halfway ...
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1answer
22 views

For the function $z = x\tan^{-1} \frac{y}{x} + y\sin^{-1} \frac{x}{y} + 2 $

For the function $$z = x\tan^{-1} \frac{y}{x} + y\sin^{-1} \frac{x}{y} + 2 $$ To find the value of $$ x\frac {\partial z}{\partial x} + y\frac {\partial z}{\partial y} $$ at (1,1) I wanted to ...
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1answer
22 views

Show $f(x,y) = (x^2+y^2)\sin(1/\sqrt{x^2+y^2}$ is differentiable at the origin

With $f(x,y) = \\(x^2+y^2)\sin(1/\sqrt{x^2+y^2} : (x,y) \neq 0, \\0 : (x,y) = 0$ Using the definition of differentiability, would I expand $f(v + h)$ (the vector representation) to f(x+h,y+h), then ...
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1answer
24 views

Find the parametrization of the curve resulting from intersection of two surfaces

The question reads as follows: Find a parametrization of the curve resulting from the intersection of the surfaces: $z = x^2 - y^2$ and $z= x^2 +xy - 1$ My attempt: (Use y = t as a parameter, so ...
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3answers
24 views

Verify second order Cauchy Riemann equations

How do I differentiate the equations in 12? I understand the hint, but I'm not sure how to act on it.
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2answers
44 views

Multivariable calculus, level curves

Let $$f(x,y) = 3x^2 + 4xy + 3y^2$$ for all $(x,y)$ in $R^2$. To study f it can be useful to implement the variable substitution $$u = x + y \qquad \textrm{and} \qquad v= x-y .$$ a) Sketch some ...
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2answers
30 views

Stuck on integration by parts question

Here's what I have so far, but I'm stuck with what to do after this
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1answer
23 views

Flux of $(0,2y,z)$ over the cylinder (?) $y=\ln(x)$

Let $S$ be the portion of the cylinder $y=\ln(x)$ (what, this is a cylinder?) in the first octant such that the projector parallel to $y$ over the plane $xz$ is the rectangle $1\le x\le e$, $0\le z\le ...
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1answer
27 views

Matrix with continuous entries

If $S$ is a metric space, if $a_{11},..,a_{mn}$ are real continuous functions on $S$ and if, for each $p\in S$, $A_p$ is the linear transformation of $\mathbb{R}^n$ into $\mathbb{R}^m$ whose matrix ...
1
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1answer
24 views

Parameterising a curve correctly

Question Parameterise the curve formed by the intersection of the sphere $x^2+y^2+z^2=1$, the cylinder $x^2+y^2=x$ and the halfspace $z \gt 0$ My attempt: The intersection between these curves is ...
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1answer
70 views

Computing the Inverse of a Fourth Order Tensor

Suppose that we have a fourth order tensor ${\bf{A}}$ $${\bf{A}}=A_{ijkl} {\bf{e}}_i \otimes {\bf{e}}_j \otimes {\bf{e}}_k \otimes {\bf{e}}_l$$ in the orthonormal basis ...
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2answers
34 views

Line integral $= 0$ meaning?

Question: Compute the line integral for the vector field $F(x,y) = (x^2y,y^2x)$ and the path $$r(t) = (\cos t,\sin t),\quad t \in [0,2\pi]$$ Answer: The answer I am getting is $0$, I am ...
2
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3answers
36 views

Example 9.18 from PMA Rudin

We know that $\gamma: (a,b)\to E\subset \mathbb{R}^n$ and $f:E\to \mathbb{R}^1$. Hence $f'(\gamma(t))\in L(\mathbb{R}^1, \mathbb{R}^1)$ since to any point $t\in(a,b)$ it corresponds some real ...
2
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0answers
24 views

Differentiability of Parameterized Inverse

Suppose I have a function $f(x,y)$ that is continuously differentiable and that for each $x$, the function $f(x,\cdot)$ is strictly increasing. Let $f_x^{-1}(\cdot)$ be the inverse function of ...
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1answer
16 views

Taking the derivative of a function of a convex combination of vectors, $f((1-t)x + t\cdot y)$

Let $f$ be a differentiable function, $x\not = y$ and vectors (say in $\mathbb{R}^n)$, and define $g:(0,1] \to \mathbb{R}$ by $$ g(t) = f((1-t)x + t\cdot y) $$ How would I differentiate this with ...
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0answers
28 views

Flux of $\vec{F}(x,y,z) = yx^2\vec{i}-2\vec{j}+3\vec{k}$ over a rectangle in $\mathbb{R}^3$

So I need to integrate: $$\vec{F}(x,y,z) = yx^2\vec{i}-2\vec{j}+3\vec{k}$$ over the region: $$y=0, -1\le x\le 2, 2\le z\le 7$$ Considering the normal vector in the direction of $-\vec{j}$ which ...
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0answers
45 views

How do I solve for the volume of a hyperboloid using a double integral in polar coordinates?

Here is the problem text, with my attempts at solving it at the bottom: Suppose you are part of a team designing a water tank in the shape of a hyperboloid. The tank is to have a top radius a of 2 ...
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0answers
21 views

Limits of the power of the double integral for finding volumes

I know that the double integral can be used to find volumes. The triple integral can also find volumes. Is the double integral limited to being only applicable for finding special cases of volumes or ...
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1answer
29 views

Trivial questions/examples about derivatives in higher dimensions

I am just learning to work with derivatives in higher dimensions. I am struggling to catch up in my new math class and I don't know where to start in tackling the following problems. A worked example ...
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32 views

The volume of null-set

I saw this claim: if $v(E)$ = 0, then E is a null-set. The converse statement is wrong. where $v(E)$ is the Jordan measure of $E$, I am not sure if this right, because I know $\int_E1_E=0 ...
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0answers
37 views

Surface integral of $z=x^2+y^2$ when enclosed by $z=2$ and $z=6$

I'm trying to take the surface integral of $$z=x^2+y^2$$ inside the region defined by $z=2$ and $z=6$ It can be proven that this is the same as integrating: ...
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0answers
23 views

Is $e^{ixt}/[i\tan(x/2)+0.5]$ complex-differentiable even when $t \in \mathbb{C}$, not just $\mathbb{R}$?

Is $$\frac{e^{ixt}}{i\tan(x/2)+0.5}$$ complex-differentiable at $t=0$ when $t \in \mathbb{C}$, not just $\mathbb{R}$? If so or not, how does one prove this? (Here, $x \in \mathbb{R}$ is some ...
2
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3answers
51 views

Difference between these two propositions

In my multivariable calculus course they ask me to prove the following propositions: 1) $lim_{(x,y)\to(0,0)}f(x,y) = L \Leftrightarrow \forall\epsilon>0, \exists\delta>0$ such that ...
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0answers
18 views

exhaustion of $[0,1]/ \Bbb{Q}$ by Jordan sets Ωm

I tried to prove that $[0,1]/ \Bbb{Q}$ can't be exhausted by jordan sets all i get is $\Omega = [0,1]/ \Bbb{Q}$ and ∂$\Omega=[0,1]$ I not sure how to continue from here. thanks ahead
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3answers
45 views

Derivative of linear transformation with confusing moment

After reading this part of Rudin's book i have one question: $A'(\mathbf{x})=A$ seems to me little bit weird because: 1) $A'(x)$ - it's derivative of operator $A$ at point $\mathbf{x}\in ...
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0answers
9 views

An algebraic equation system and the Jacobi determinant as test for its solvability

I am trying to verify a result in a text that I am currently reading. The context is in algebra and combinatorics. However the result is obtained by using a bit of vector calculus much to my suprise. ...
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1answer
102 views

How to prove that $\lim_{(x, y) \to (0, 0)} \left(\frac{1}{|x|} + \frac{1}{|y|}\right) = 0$? [closed]

Prove that $$\lim_{(x, y) \to (0, 0)} \left(\frac{1}{|x|} + \frac{1}{|y|}\right) = 0$$ I couldn't prove this. Please suggest a solution.
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1answer
45 views

Is $g$ is differentiable?

Let $f:\Bbb R\to \Bbb R$ be a differentiable function. Define $g:\Bbb R^2\to \Bbb R $ as $g(x,y) =f(\sqrt {x^2+y^2})$. Is $g$ differentiable? If we can show that one of the partial derivatives ...
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0answers
9 views

$ \vec{F}=<x^{x^2},(x+z)\sin(y^3),(y^2-x^2+2yz)>$;C is the boundary of the triangular part of the plane $x+y+z=3$ that lies in the first octant

I have a question which is following; $\vec{F}=\langle x^{x^2},(x+z)\sin(y^3),(y^2-x^2+2yz)\rangle$;C is the boundary of the triangular part of the plane $x+y+z=3$ that lies in the first octant. I ...
2
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0answers
21 views

If function differentiable in $E\subset \mathbb{R}^n$ then it's continuous on it

Let $E$ is an open subset in $\mathbb{R}^n,$ $\mathbf{f}$ maps $E$ into $\mathbb{R}^m$. Prove that $\mathbf{f}$ is continuous at any point of E at which $\mathbf{f}$ is differentiable. Proof: Let ...
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0answers
23 views

Uniqueness of derivative in $\mathbb{R}^n$

There is one confusing moment. How did Rudin get (16)? My though is the following: Since $\dfrac{|B\mathbf{h}|}{|\mathbf{h}|}\to 0$ as $\mathbf{h}\to \mathbf{0}$. Fixing $\mathbf{h}\neq\mathbf{0}$ ...