Let F⃗(x,y,z)=7x^2i⃗−sin(xy)(i⃗+j⃗).
Calculate the divergence: div F→(x,y,z)=
To calculate the divergence of a vector field, we will need to compute the divergence of each component and add them up.
Let's start by calculating the divergence of the first component, 7x^2i⃗.
The divergence of a vector field in terms of its x-component is given by ∂F_x/∂x, where F_x is the x-component of the vector field.
Taking the derivative of 7x^2 with respect to x, we get:
∂(7x^2)/∂x = 14x
So, the divergence of the x-component is 14x.
Next, let's calculate the divergence of the second component, -sin(xy)(i⃗+j⃗).
The divergence of a vector field in terms of its y-component is given by ∂F_y/∂y, where F_y is the y-component of the vector field.
Taking the derivative of -sin(xy) with respect to y, we get:
∂(-sin(xy))/∂y = -x*cos(xy)
So, the divergence of the y-component is -x*cos(xy).
Finally, let's calculate the divergence of the third component, which is 0 since there is no z-component in the vector field.
Now, let's add up the three components:
div F→(x,y,z) = 14x + (-x*cos(xy)) + 0
Simplifying, we have:
div F→(x,y,z) = 14x - x*cos(xy)
Therefore, the divergence of F→(x,y,z) is 14x - x*cos(xy).
To calculate the divergence of a vector field F→(x,y,z) = 7x^2i⃗ -sin(xy)(i⃗+j⃗), you can use the divergence operator ∇·F→, which is defined as the dot product of the gradient operator ∇ and the vector field F→.
The gradient operator ∇ is defined as:
∇ = ∂/∂x i⃗ + ∂/∂y j⃗ + ∂/∂z k⃗
So, in this case, the gradient of F→ is:
∇F→ = (∂/∂x, ∂/∂y, ∂/∂z) (7x^2i⃗ -sin(xy)(i⃗+j⃗))
Now, we can find the divergence by taking the dot product of the gradient operator and the vector field:
div F→ = ∇·F→ = (∂/∂x, ∂/∂y, ∂/∂z) · (7x^2i⃗ -sin(xy)(i⃗+j⃗))
To calculate the dot product, we need to distribute and combine like terms:
div F→ = (∂/∂x (7x^2) + ∂/∂y (-sin(xy)) + ∂/∂z 0)i⃗ + (∂/∂x 0 + ∂/∂y (-sin(xy)) + ∂/∂z 0)j⃗ + (∂/∂x 0 + ∂/∂y 0 + ∂/∂z 0)k⃗
Since the z-component of F→ is zero, the divergence only depends on the x and y components:
div F→ = ∂/∂x (7x^2) + ∂/∂y (-sin(xy))
Now, we can find the partial derivatives:
∂/∂x (7x^2) = 14x
∂/∂y (-sin(xy)) = -x cos(xy)
Therefore, the divergence of F→(x,y,z) is given by:
div F→ = 14x - x cos(xy)
Hmmm. No k component?
Oh, well. Combining the terms,
F(x,y,z) = (7x^2 - sin(xy))i - sin(xy)j + 0k
So,
div F = ∇•F
= (14x-ysin(xy))i - xsin(xy)j + 0k