2 votes
Accepted

Double Integrals - Region delimited by triangle in counterclockwise

By green’s theorem, we can convert our line integral (which would require 3 integrals to solve) into a single double integral. So far, your setup is correct, where we have $$\int_{\Omega}\left[e^{x+y} ...
1 vote

How to approach and resolve this problem?

The partial derivative means that you take the derivative of the functions to all its variables on which is depends explicitly. As an example, say we have $k(x) = x^2$, with $x(t) = 4t$. The partial ...
1 vote

Proving $\frac{dr}{dx_i}$=$\frac{x_i}{r}$ in Index Notation

I think that with $r^2$ you mean $r^2=x_1^2+x_2^2+x_3^2$. Appart from that, the 0 gives you problems, but for any other point just calculate the classic derivative of $\sqrt{x_1^2+x_2^2+x_3^2}$ with ...
1 vote
Accepted

Lagrange multiplier $f(x,y)=x^2+2xy^2+y^2$. Show that $(-1,1)$ is the minimal point of $f$.

The solution (-1,1) is not a global min. The problem has no global min, it is unbounded to $-\infty$. To see that make the substitution $x=1-2y$ in the objective function to have now an unconstrained ...
  • 3,782
1 vote
Accepted

Can someone help me understand the difference between gradient vector and directional derivative?

The gradient is the direction of maximum ascent, as you pointed out, in the underlying domain of the function. The directional derivative, on the other hand, is the rate of change in the function, in ...

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