Vector Analysis ifferential Calculus

So, in summary, the height of a certain hill is given by the function h(x,y) = 10(2xy-3x^2-4y^2-18x+28y+12), where x is the distance east of South Hadley and y is the distance north. To find the location of the top of the hill, we need to take the partial derivatives of h(x,y) with respect to x and y, and set them equal to 0. After correcting some errors in the calculation of the partial derivatives, we get the equations 20y-90x-180=0 and 20x-80y+280=0. Solving these equations gives us the values for x and
  • #1
Benzoate
422
0
Vector Analysis:Differential Calculus

Homework Statement



The height of a certain hill(in feet) is given by

h(x,y)=10(2xy-3x^2-4y^2-18x+28y+12)

where y is the distance (in miles) north, x the distance east of South Hadley

a)Where is the top of the hill located

b) How high is the hill?

Homework Equations



grad T=dT/dx xhat+dT/dy yhat+ dT/dz zhat

The Attempt at a Solution



a) I need to find the distance in the x direction , so I would take the derivative of h(x,y) with respect to x

dh/dx=20*x-12=0=> x=3/5 feet

b) same algorithm, only I am now ask to calculate how high the hill is and so I would take the derivative of h(x,y) with respect to y:

dh/dy=y=3*x+9=3*(.6)+9=10.8 feet

or maybe I should calculate h(x,y) in order to determine the height of the hill. Therefore , I'd plugged x and y into h(x,y) right?
 
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  • #2
Hi Benzoate! :smile:

(have a curly d: ∂ :smile:)
Benzoate said:
The height of a certain hill(in feet) is given by

h(x,y)=10(2xy-3x^2-4y^2-18x+28y+12)
…
a) I need to find the distance in the x direction , so I would take the derivative of h(x,y) with respect to x

dh/dx=20*x-12=0=> x=3/5 feet

b) same algorithm, only I am now ask to calculate how high the hill is and so I would take the derivative of h(x,y) with respect to y:

dh/dy=y=3*x+9=3*(.6)+9=10.8 feet

Nooo … :cry:

Your ∂h/∂x and ∂h/∂x are completely wrong …

for example, ∂h/∂x should start with 20*y, not 20*x

and what happened to all the other terms (and all the other 10s)?

You need to go back to your book and look again at how to do partial derivatives … :smile:
 
  • #3


Benzoate said:

Homework Statement



The height of a certain hill(in feet) is given by

h(x,y)=10(2xy-3x^2-4y^2-18x+28y+12)

where y is the distance (in miles) north, x the distance east of South Hadley

a)Where is the top of the hill located

b) How high is the hill?




Homework Equations



grad T=dT/dx xhat+dT/dy yhat+ dT/dz zhat

The Attempt at a Solution



a) I need to find the distance in the x direction , so I would take the derivative of h(x,y) with respect to x

dh/dx=20*x-12=0=> x=3/5 feet

b) same algorithm, only I am now ask to calculate how high the hill is and so I would take the derivative of h(x,y) with respect to y:

dh/dy=y=3*x+9=3*(.6)+9=10.8 feet

or maybe I should calculate h(x,y) in order to determine the height of the hill. Therefore , I'd plugged x and y into h(x,y) right?

wow I messed up big time with calculating my partial derivatives.

Anyway, dh/dx= 20y-90x-180= and dh/dy= 20x-80y+280 =0. once I calculate my values for x and y , I would be able to calculate the height which is h(x,y), correct?
 
  • #4
Benzoate said:
Anyway, dh/dx= 20y-90x-180= and dh/dy= 20x-80y+280 =0.

(what happened to that ∂ I gave you? :smile:)

erm … 90x is wrong :rolleyes:

and the equations would be a lot more manageable if you'd divided them by 20 :wink:
once I calculate my values for x and y , I would be able to calculate the height which is h(x,y), correct?

That's right! :smile:
 
  • #5


Why is this listed under "physics" rather than "mathematics"?
 
  • #6


HallsofIvy said:
Why is this listed under "physics" rather than "mathematics"?

well because the problem came from my intro to Electrodynamics textbook
 
  • #7
it's field theory …

Benzoate said:
well because the problem came from my intro to Electrodynamics textbook
Benzoate said:
The height of a certain hill …​

:biggrin: The hill was part of a field! :biggrin:​
 

Related to Vector Analysis ifferential Calculus

1. What is vector analysis in differential calculus?

Vector analysis in differential calculus is a mathematical tool used to analyze and manipulate vectors (quantities with magnitude and direction) in a multi-dimensional space. It involves the use of differential operators, such as gradient, divergence, and curl, to study the behavior of vector fields.

2. How is vector analysis different from scalar analysis?

The main difference between vector analysis and scalar analysis is that vector analysis deals with quantities that have both magnitude and direction, while scalar analysis deals with quantities that only have magnitude. Vector analysis also involves the use of differential operators, while scalar analysis does not.

3. What are some applications of vector analysis in differential calculus?

Vector analysis in differential calculus has numerous applications in physics, engineering, and other fields. It is used to study fluid flow, electromagnetic fields, and motion in three-dimensional space. It is also used in optimization problems, such as finding the shortest distance between two points.

4. What are some common vector operations used in vector analysis?

Some common vector operations used in vector analysis include dot product, cross product, and vector differentiation. The dot product is used to find the angle between two vectors or to project one vector onto another. The cross product is used to find a vector perpendicular to two given vectors. Vector differentiation is used to find the rate of change of a vector field at a certain point.

5. What are some important theorems in vector analysis?

Some important theorems in vector analysis include the gradient theorem, divergence theorem, and Stokes' theorem. These theorems relate the behavior of vector fields to the behavior of their differential operators. They are used to solve various types of problems, such as finding the work done by a force field or the flux of a vector field through a closed surface.

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