# What is a gradent

1. Sep 8, 2008

### thharrimw

I am a high school senior and I taught myself calc AB and got a 4 on the test last year but now I'm in my high school's calc class and i already know everything that is in the couse so I'm perparing to take the BC test and in my calc book it dosn't give a good explanation of what a gradent is or where it is used so firts of all what is a gradent? and then are they on the BC calc test? where are they used?

2. Sep 8, 2008

### rbeale98

without getting into formulae, the general concept is that a gradient is any kind of slope. slope of a line is a gradient. it is commonly applied to more complex relationships. in engineering it is commonly used to describe the rate of change of some parameter (temperature, pressure) over a distance. also, it is easily viewed on a topographic map when you have high gradients. I'll let others get into the more complex discussion.

3. Sep 8, 2008

### thharrimw

so a gradent is simmler to a deritive.

4. Sep 8, 2008

### rbeale98

it is a form of the derivitive. if you calculate the gradient for a function f(x,y,z) (a 3-d surface) you basically get a vector in terms of the unit vectors i,j,k and variables x,y,z. when you plug in a point (x,y,z) on the surface into the gradient vector, you get a vector that points in the direction of steepest ascent and the magnitude of the vector is the value of the slope of the surface in that direction at that point (x,y,z).

5. Sep 8, 2008

### thharrimw

so the partial deritives that i've been learning how to find are used to find gradients?

6. Sep 8, 2008

### rbeale98

yes. and to correct myself above a 3-d surface would be z=f(x,y) not f(x,y,z)

7. Sep 9, 2008

### thharrimw

how do you use partial deritives to find a gradient? then what are gradents used for (pratical applacations)?

8. Sep 9, 2008

9. Sep 9, 2008

### HallsofIvy

Staff Emeritus
In England, it is common to refer to the derivative of a function of a single variable as the 'graident'. In the United States, that is normally reserved for functions of several variables. If f is a function of x, y,z, the "gradient" of f, sometimes abbreviated grad f, sometimes symbolized as $\nabla f$, is the vector each of whose components is a partial derivative of f: $<\partial f/\partial x, \partial f/\partial y, \partial f/\partial z>$. You are correct in that it is the derivative. It is possible for a function which is NOT differentiable at a point to have partial derivatives there but the gradient cannot exist where the function is not differentiable.

If f(x,y,z) is differentiable at a point, then it has a derivative in any direction- and the derivative in the direction of the unit vector $\vec{v}$ is $\nabla f\cdot\vect{v}$

The gradient of f always points in the direction of fastest increase (the derivative is largest) and its length is that largest derivative.

10. Sep 9, 2008

### thharrimw

so you add up all the partials to find the gradient of a function?
do i need to know how to use poler equations?
gradient is just another word for slope of grade?

11. Sep 9, 2008

### NoMoreExams

Where do you see anything about adding up the partials? Gradient is a vector thus it has components, is that what you are refering to? When you do "add up partials", are you thinking of divergence? In which case you have $$\nabla \cdot \mathbf{F} = \frac{\partial \mathbf{F}}{\partial x} + \frac{\partial \mathbf{F} }{\partial y} + \frac{\partial \mathbf{F}}{\partial z}$$ assuming $$\mathbf{F} = F_{x} \mathbf{i} + F_{y} \mathbf{j}+ F_{z} \mathbf{k}$$ is a cont. diff. vector field

Last edited: Sep 9, 2008
12. Sep 9, 2008

### thharrimw

yah so
[tex] \nabla \cdot \mathbf{F} = \frac{\partial \mathbf{F}}{\partial x} + \frac{\partial \mathbf{F} }{\partial y} + \frac{\partial \mathbf{F}}{\partial z} [tex]
is just stating that each parital is a just one component of a vector........ i wasn't thinking about a 3D vector....... I am still getting used to dealing with 3D equations.so for a 2D vector you could write it as [tex] \nabla \cdot \mathbf{F} = \frac{\partial \mathbf{F}}{\partial x} + \frac{\partial \mathbf{F} }{\partial y} [tex]

13. Sep 9, 2008

### thharrimw

sory i don't know how to use LaTeX but Grad F(x)= dF/dx+dF/dy
Right?

14. Sep 9, 2008

### NoMoreExams

No, that's divergence i.e. the grad operator dotted with F where "dotted" means dot product.

15. Sep 9, 2008

### thharrimw

how do you find a dot product and what is a dot product?

16. Sep 9, 2008

### malawi_glenn

17. Sep 9, 2008

### thharrimw

i've done vectors but only in physics. i've herd of dot product but i only have a book and the internet to learn from and i haden't heard of it untill now. I am alredy over my high school calc teachers head. she can't do anything with 3D.I'm also working on diffy Q's right now and basic limits for my actual calc class. I don't know what i should know or where to go next.

18. Sep 9, 2008

### thharrimw

19. Sep 9, 2008