# Tagged Questions

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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### finding bounds of parametric variables

Compute the area of the surface $$x^2+y^2-z^2=2y+2z$$ where $-1\leq z \leq 0$ You can get it in the form $x^2+(y-1)^2=(z+1)^2$ I parametrised it as $r(u,v)=(u\cos v, u\sin v+1, u-1)$. I know that ...
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### Partial Derivative of a outer product in Vector Calculus

I am trying to compute the partial derivative of certain vector products for calculating the stiffness matrix. So we already know that For any vector $\textbf{x}$, we have 1) The derivative of the ...
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### Prove that a multivariable function has a global minimum

I'm doing an Introduction to Machine Learning course by myself using some open university coursebook and it has the following question which I've tried to solve, but to no avail: Let there be a ...
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### Continunity of a two variable function in Apostol's analysis

In discussing how the concept of differentiability implying continuity cannot be applied to functions of several variables, Apostol proceeds to give an example to demonstrate why. The function he uses ...
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### prove that $f(x,y)= \frac{x^3y}{x^4+y^2}$ for $(x,y) \neq (0,0)$ and $f(0,0) = 0$ is continous at $(0,0)$ [duplicate]

How to prove that $f: \mathbb R^2 \to \mathbb R$ $f(x,y)= \frac{x^3y}{x^4+y^2}$ for $(x,y) \neq (0,0)$ and $f(0,0) = 0$ is continous at $(0,0)$? By definition: Let $\epsilon>0$ I need to prove ...
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### Prove $f(x,y) = \frac{x^2+y^2}{x+y}$ is not continuous at $(0,0)$.

Let $f(x,y) = \frac{x^2+y^2}{x+y}$ when $x+y \neq 0$ and $f(x,y) = 0$ when $x+y=0$. Prove $f$ is not continuous at $(0,0)$ in the $R^2$ norm. Is this as easy as noticing that the function is ...
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### Evaluate the following spherical coordinate triple integral

Evaluate the following \begin{align*} \int_{0}^{\pi} \int_{0}^{\pi/3} \int_{\sec \phi}^{2} 5\rho^2 \sin(\phi) \ d\rho \ d\phi \ d\theta \end{align*} Attempt at solution: We have \begin{align*} 5 \...
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### If $f_{xy}$ , $f_{yx}$ are continuous at $(x_{0},y_{0})$,then $f_{x},f_{y}$ are continuous at $(x_{0},y_{0})$?
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ The second edition Let $f$ be a function of two variables,let$(x_{0},y_{0})$ be a point and let $U$ be an open disk with center $(x_{0},y_{0})$.Assume ...