Question

A Welche Form haben die Niveauflächen der skalaren Felder a) \(\varphi(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{\frac{3}{2}}\) b) \(\varphi(x, y, z)=x^{2}+y^{2}\) c) \(\varphi(x, y, z)=x+y-3 z\) B Berechnen Sie die Gradienten für diese drei skalaren Felder.

Short answer

a) The form of the level surfaces: a sphere with radius \(c\) for \(\varphi(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{\frac{3}{2}}\), a circle in the xy plane for \(\varphi(x, y, z)=x^{2}+y^{2}\), and a plane for \(\varphi(x, y, z)=x+y-3z\). b) The gradients: \(\nabla \varphi = 3x\sqrt{x^2 + y^2 + z^2} \vec{i} + 3y\sqrt{x^2 + y^2 + z^2} \vec{j} + 3z\sqrt{x^2 + y^2 + z^2} \vec{k}\) for \(\varphi(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{\frac{3}{2}}\), \(\nabla \varphi = 2x \vec{i} + 2y \vec{j}\) for \(\varphi(x, y, z)=x^{2}+y^{2}\), and \(\nabla \varphi = \vec{i} + \vec{j} - 3\vec{k}\) for \(\varphi(x, y, z)=x+y-3z\).

Step-by-step solution

Find forms of level surfaces for scalar fields

a) A possible level surface of the scalar field \(\varphi(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{\frac{3}{2}}\) can be found by setting the function equal to a constant, say \(c^3\), so that \(x^{2}+y^{2}+z^{2} = c^2\). This is the equation of a sphere with radius \(c\) in 3D.b) For \(\varphi(x, y, z)=x^{2}+y^{2}\), a possible level surface would be the one when \(\varphi\) equals a constant, say \(d\), to give \(x^{2}+y^{2} = d\). This represents a circle in the xy plane.c) for \(\varphi(x, y, z)=x+y-3z\), setting this equal to a constant \(m\) gives a level surface defined by the plane \(x+y-3z = m\).

Calculate the gradients

a) The gradient of \(\varphi(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{\frac{3}{2}}\) is calculated by taking the partial derivatives of the function with respect to \(x, y, z\) respectively. The gradient vector is \(\nabla \varphi = 3x\sqrt{x^2 + y^2 + z^2} \vec{i} + 3y\sqrt{x^2 + y^2 + z^2} \vec{j} + 3z\sqrt{x^2 + y^2 + z^2} \vec{k}\).b) Similarly, for \(\varphi(x, y, z)=x^{2}+y^{2}\), the gradient vector is \(\nabla \varphi = 2x \vec{i} + 2y \vec{j}\).c) For \(\varphi(x, y, z)=x+y-3z\), the gradient \(\nabla \varphi = \vec{i} + \vec{j} - 3\vec{k}\).

Key concepts

Niveauflächen

Imagine taking a hike on a mountain and wanting to stay at the same altitude no matter where you go. The path you follow on the mountain represents a 'Niveaufläche', or a level surface in physics. These surfaces are invisible boundaries where a scalar field, like gravitational potential or temperature, remains constant.

Let's consider the scalar fields from the exercise. For the field \(\phi(x, y, z) = (x^2 + y^2 + z^2)^{\frac{3}{2}}\), the Niveauflächen are spheres. As you stand on one of these spheres and walk around, your distance from the origin never changes, just like your altitude would remain the same while walking along a contour line on a mountain.

In the second scenario, \(\phi(x, y, z) = x^2 + y^2\), we see a 2D version of level surfaces—circles in the xy-plane. It's like looking at the contour lines of a mountain from above, each representing a specific altitude. Lastly, \(\phi(x, y, z) = x + y - 3z\) gives us level planes, slicing through the three-dimensional space at an angle, describing a constant height above a tilted reference plane.

Gradient of a Scalar Field

Ever stood at the bottom of a hill feeling a breeze coming straight down the slope? That breeze is somewhat like the gradient of a scalar field, a vector pointing in the direction of the steepest increase of the field's value. It's as if the gradient is telling you, 'Climb this way if you want to go up fastest!'

For \(\phi(x, y, z) = (x^2 + y^2 + z^2)^{\frac{3}{2}}\), the gradient points radially outward, meaning if you were a tiny ant standing on the spherical Niveaufläche, the gradient would be like a finger pushing you directly away from the center. Similarly, for a field like \(\phi(x, y, z) = x^2 + y^2\), the gradient vector lays flat on the xy-plane, urging you to move perpendicular to the circles.

In the case of the inclined plane defined by \(\phi(x, y, z) = x + y - 3z\), the gradient vector orients parallel to the xy-plane, showing the direction to move within the plane for the fastest increase in field value.

Partial Derivatives

Partial derivatives are like detectives examining how a function changes when you tweak just one variable at a time while keeping others constant. Picture adjusting the brightness on your phone screen. Tinkering with the brightness would be analogous to taking a partial derivative with respect to the brightness setting.

When looking at a function such as \(\phi(x, y, z) = (x^2 + y^2 + z^2)^{\frac{3}{2}}\), taking the partial derivative with respect to \(x\) means observing how the function changes if \(x\) alone varies, while \(y\) and \(z\) are snoozing. You apply the same idea to the function \(\phi(x, y, z) = x^2 + y^2\), treating \(x\) and \(y\) as solo players in turn.

Likewise, with \(\phi(x, y, z) = x + y - 3z\), the partial derivatives give us the change in \(\phi\) due to tiny shifts in \(x\), \(y\), or \(z\) independently. These partial derivatives come together to form the gradient vector, which is key to visualizing how the field changes in space.