The Quantity Z Can Be Expressed as a Function of X

question 33

Essay

The quantity z can be expressed as a function of x and y as follows: z = f(x, y).Now x and y are themselves functions of r and $\theta$ , as follows: $x = g ( r , \theta )$ and $y = h ( r , \theta )$ Suppose you know that g(1, $\pi$ /2)= -1, and h(1, $\pi$ /2)= 1.In addition, you are told that $$\begin{array} { l } \frac { \partial f } { \partial x } ( - 1,1 ) = 1 , \quad \frac { \partial f } { \partial y } ( - 1,1 ) = 6 , \quad \frac { \partial g } { \partial r } \left( 1 , \frac { \pi } { 2 } \right) = 7 \\\frac { \partial g } { \partial \theta } \left( 1 , \frac { \pi } { 2 } \right) = 7 , \quad \frac { \partial h } { \partial r } \left( 1 , \frac { \pi } { 2 } \right) = 6 , \quad \frac { \partial h } { \partial \theta } \left( 1 , \frac { \pi } { 2 } \right) = 4\end{array}$$ Find $\frac { \partial z } { \partial r } ( 1 , \pi / 2 )$

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Definitions:

Franz Anton Mesmer

An 18th-century German physician who is best known for his theory of animal magnetism and is considered a pioneer in the field of hypnotism.

Cerebral Cortex

The outer layer of the cerebrum in the brain, playing a key role in memory, attention, perception, cognition, awareness, thought, language, and consciousness.

Animal Magnetism

A historical concept that hypothesized a natural etheric fluid thought to exist in the human body, related to the now-debunked science of mesmerism.

Hypnotized

A state in which a person appears to be in a trance, often induced by another person, where they can be more easily influenced by suggestions.