Implicit Differentiation and Partial Derivatives
We are given an implicit function $G(x, y, z) = 0$. We need to find the value of the product $\left(\frac{\partial z}{\partial x} \times \frac{\partial x}{\partial y} \times \frac{\partial y}{\partial z}\right)$. We assume $G$'s partial derivatives exist and are non-zero where needed.
Finding Partial Derivatives
Using the rules of implicit differentiation:
- To find $\frac{\partial z}{\partial x}$, we treat $y$ as a constant and differentiate $G$ with respect to $x$, considering $z$ as a function of $x$ and $y$:
$ \frac{\partial G}{\partial x} + \frac{\partial G}{\partial z} \frac{\partial z}{\partial x} = 0 $
Solving for $\frac{\partial z}{\partial x}$ yields:
$ \frac{\partial z}{\partial x} = -\frac{\frac{\partial G}{\partial x}}{\frac{\partial G}{\partial z}} $
- To find $\frac{\partial x}{\partial y}$, we treat $z$ as a constant and differentiate $G$ with respect to $y$, considering $x$ as a function of $y$ and $z$:
$ \frac{\partial G}{\partial y} + \frac{\partial G}{\partial x} \frac{\partial x}{\partial y} = 0 $
Solving for $\frac{\partial x}{\partial y}$ yields:
$ \frac{\partial x}{\partial y} = -\frac{\frac{\partial G}{\partial y}}{\frac{\partial G}{\partial x}} $
- To find $\frac{\partial y}{\partial z}$, we treat $x$ as a constant and differentiate $G$ with respect to $z$, considering $y$ as a function of $x$ and $z$:
$ \frac{\partial G}{\partial z} + \frac{\partial G}{\partial y} \frac{\partial y}{\partial z} = 0 $
Solving for $\frac{\partial y}{\partial z}$ yields:
$ \frac{\partial y}{\partial z} = -\frac{\frac{\partial G}{\partial z}}{\frac{\partial G}{\partial y}} $
Calculating the Product
Now, we multiply these three partial derivatives:
$ \left(\frac{\partial z}{\partial x} \times \frac{\partial x}{\partial y} \times \frac{\partial y}{\partial z}\right) = \left(-\frac{\frac{\partial G}{\partial x}}{\frac{\partial G}{\partial z}}\right) \times \left(-\frac{\frac{\partial G}{\partial y}}{\frac{\partial G}{\partial x}}\right) \times \left(-\frac{\frac{\partial G}{\partial z}}{\frac{\partial G}{\partial y}}\right) $
Simplify the expression by cancelling terms:
$ = (-1) \times (-1) \times (-1) \times \left(\frac{\frac{\partial G}{\partial x}}{\frac{\partial G}{\partial x}}\right) \times \left(\frac{\frac{\partial G}{\partial y}}{\frac{\partial G}{\partial y}}\right) \times \left(\frac{\frac{\partial G}{\partial z}}{\frac{\partial G}{\partial z}}\right) $
$ = (-1) \times 1 \times 1 \times 1 $
$ = -1 $
Final Result
The value of the product $\left(\frac{\partial z}{\partial x} \times \frac{\partial x}{\partial y} \times \frac{\partial y}{\partial z}\right)$ is -1.
Important Questions from Numerical Computation
$A$ is an ($n \times n$) matrix. Consider the following two statements
Statement 1: Columns of matrix $A$ are linearly independent
Statement 2: Inverse of matrix $A$ exists
Which one of the following statements is TRUE?
- Levenshtein distance is used to measure the minimum edit distance between two strings by counting the minimum number of editing operations (such as, insertions, deletions, substitutions) required to transform one string to another. Consider an alternative version of the Levenshtein distance in which each insertion and each deletion has a cost of 1 and substitutions are not allowed.
Based on this alternative version, the Levenshtein distance between the strings, word and work is ______ (Answer in integer). - In a given collection of documents, let $N$ be the total number of documents and let $d$ be the number of documents in which the term $t$ occurs.
Which ONE of the following fractions is used to define the inverse document frequency (idf) of the term $t$? - The total number of unique character bigrams in the string thoughtsoftheghosts (which contains 19 characters) is ______.
A bacterium can be approximated as a cylinder with a hemisphere at each end, as shown in the figure. The cylinder has a height of $1 \mu\text{m}$ and diameter of $1 \mu\text{m}$. Assuming that the density of the bacterium is equal to that of water, what is the approximate mass of this bacterium?
Given: density of water = $10^3 \text{ kg/m}^3$; $1 \mu\text{m} = 10^{-6}\text{ m}$
Volume of a cylinder = $\pi r^2 h$, where $r$ is the radius and $h$ is the height of the cylinder
Volume of a sphere = $\frac{4}{3} \pi r^3$, where $r$ is the radius of the sphere
