# 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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### Modeling of Multivariate Function of Dependent Variables

In multi-variable calculus, if I write $f(x,y,z)$, it is assumed that $x,y,z$ are independent. I'd like to model a quantity $F$, that is a function of 3 related quantities, $x,y,z$. In fact, $xy=z$. ...
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### Can't Finish Double Integral in Polar or Cartesian

Alright, so I'm stuck on what I think should be a simple double integral. It is $\int_0^1\int_{\sqrt x}^1e^{y^3} \, dy \, dx$. This is just the volume between the surface $z=e^{y^3}$ and the area ...
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### Line Integral: $\int_C{x^2}\:dy$

How can I calculate $\int_C{x^2}\:dy$ in which $C$ is a line segment from the point $(0,0)$ to $(3,2)$? I am new to line integrals, I am only familiar when given a function and in $ds$. How can I do ...
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### Evaluating the line integral $\int_C{F\cdot dr}$ for a particular conservative vector field $F$

So I have this two dimensional vector field: $$F=\langle (1+xy)e^{xy},x^2e^{xy}\rangle$$ How can I tell whether $F$ is conservative or not? And also how do I calculate $\int_C{F\cdot dr}$, where $C$ ...
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### Calculating min/max of a multivariate function on a region

This video shows an example of how to find the absolute maxima and minima of the function $f=xy+y^2$ at the region $\{(x,y):|x|\leq1,|y|\leq2\}$. I understand why he set $f_x, f_y$ to $0$, checked ...
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### Green theorem application

Suppose that a simple closed curve $C$ in the $xy$ plane, that bounds a convex domain $D$ containing the origin. The curve is specified by $x=f(\varphi), y=g(\varphi)$ where $0\leq \varphi< 2\pi$ (...
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### Doesn't $x^3+2y^3+3z^3=0$ give a surface in $R^3$?

In my last exam on Advanced Calculus (following Spivak's Calculus on Manifolds), I couldn't solve the following question. True or false: the set $S$ in $R^3$ given by $x^3+2y^3+3z^3=0$ is a ...
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### Intuition behind surface integrals

While line integrals derive their intuition from , and are analogous to, the concept of Work in physics, what intuition is there to back up the notion of surface integrals? In the texts I've been ...
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### How does gradient of a vector point steepest ascent

The derivative of distance function with respect to time give velocity function in single variable calculus. But how does gradient of a multivariable function point steepest ascent? I have been ...
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### Let $\vec F(x,y)=(y+xg(x),y^2), \vec F(1,1)=(3,1)$. $\vec F_x \perp \vec F_y$.Find $g$.

Let $\vec F(x,y)=(y+xg(x),y^2), \vec F(1,1)=(3,1)$. $\vec F_x \perp \vec F_y$ Find $g$. Attempt: I look for the partial derivatives, I did so differentiating each coordinate with respect to $x$...
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### Function of several variables which is continuous at single point

Examples of functions on $\mathbb{R}$ which are continuous at a single point are well known. But what about $f:\mathbb{R}^2\to \mathbb{R}$ which is continuous at a single point? I tried to proceed as ...
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### How are these two terms in $y$ removed from the triple integral? (Divergence theorem?)

I will post the photo here for convenience sake. I wish to understand why it just says, odd in $y$ and then cancels the $y$ bits and simplifies the integral a whole lot. Here is the scan: http://i....
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### Prove that $f(v_1, v_2)$ is greater 0 $\forall v_1, v_2$
I have the function $f_{a, b, c}\colon \mathbb{R}^2 \to \mathbb{R}$, $f_{a, b, c}(v_1, v_2) = av_1^2 + 2bv_1v_2 + cv_2^2$. I want to know for which $a$, $b$ and $c$ this function \$f_{a, b, c}(v_1, ...