Yes, solve the three differential equations to find F(x,y,z) up to a constant. The given condition sets this constant, from which you can find F(x,y,z) for all x,y,z.jonroberts74 said:my book doesn't have a good example of a problem like this, am I looking for a potential?
##\frac{\partial}{\partial x} = 2xyze^{x^2} \Rightarrow \int 2xyze^{x^2}dx = yze^{x^2} + h(y,z)##jonroberts74 said:Homework Statement
##\nabla{F} = <2xyze^{x^2},ze^{x^2},ye^{x^2}##
if f(0,0,0) = 5 find f(1,1,2)Homework Equations
The Attempt at a Solution
my book doesn't have a good example of a problem like this, am I looking for a potential?
##<\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z} >= <2xyze^{x^2},ze^{x^2},ye^{x^2}>##
jonroberts74 said:##\frac{\partial}{\partial x} = 2xyze^{x^2} \Rightarrow \int 2xyze^{x^2}dx = yze^{x^2} + h(y,z)##
##\frac{\partial}{\partial y} = ze^{x^2} = ze^{x^2} + h(y,z) \Rightarrow h_{y}(y,z) = 0##
##\frac{\partial}{\partial z} = ye^{x^2} = ye^{x^2}+h(z) \Rightarrow h_{z}(z) = h'(z)##
##h'(z) = 0 \Rightarrow h(z) = k## for a constant k, and because f(0,0,0) = 5 then
##f= yze^{x^2} + 5##
so ##f(1,1,2) = 1(2)e^1+5 = 2e+5##
A line integral is a mathematical concept used in vector calculus to find the total value of a function along a curve or path. It takes into account not only the magnitude of the function, but also its direction along the path.
A gradient field is a vector field that represents the rate of change of a scalar field in all directions. It is typically represented by a set of arrows pointing in the direction of the steepest increase of the scalar field.
Line integrals and gradient fields are related through the fundamental theorem of calculus, which states that the line integral of a gradient field over a curve is equal to the difference in values of the scalar field at the endpoints of the curve.
In physics, line integrals and gradient fields are used to calculate work, potential energy, and electric and magnetic fields. They are also used to describe the flow of fields through a surface or volume, which is important in understanding fluid dynamics and electromagnetism.
Line integrals and gradient fields can be calculated using various methods, including the fundamental theorem of calculus, Green's theorem, and Stokes' theorem. These theorems provide formulas for calculating line integrals and gradient fields for different types of curves and surfaces.