 It was mentioned earlier that stress is a vector, and we know that vectors can be broken down into components. So let's examine this concept in detail. Take the generalized body shown here in blue subjected to several forces, f1, f2, f3, and f4. If we take a cross-section through this body, we know that there has to be some sort of stress distribution acting on the cross-sectional area to maintain equilibrium. If we now look at a small area, delta A of this surface, we know that there is a small resultant delta F acting on that area. If we zoom in on that and then apply a coordinate system to that surface where z is normal to the surface and x and y are parallel to the surface, we can see that the resultant force delta F can be broken down into a normal component delta Fz and two components parallel to the surface delta Fx and delta Fy. These, however, are force components, not stress components. We can, however, convert them into stresses as we did before. If we look perpendicular to the surface, we can define a normal stress, and that normal stress is the limit of the resultant component of force delta Fz divided by the area delta A in the limit as delta A approaches zero. We will call this stress sigma zz. The first subscript in this labeling refers to the outward surface normal direction, so the normal vector for our surface delta A points in the z direction. The second subscript refers to the resultant force direction, so delta Fz is also in the z direction. Now for a normal stress, the resultant force direction and the outward surface normal direction will always be the same, so we often write this as just sigma z. Parallel to the surface, we get two stress components and we will call these shear stresses. The first one we will denote by the Greek letter tau and it is the resultant shear stress in the x direction, so that is the limit of delta Fx divided by delta A as delta A approaches zero. We use the same subscript notation here, so the first subscript z refers to the outward surface normal direction and the second subscript refers to the resultant force direction, which is in the x direction. Similarly, we get a second shear stress component that the resultant acts in the y direction. We call this tau zy and it is the limit of delta Fy over delta A as delta A approaches zero. Let's now look at the general state of stress in three dimensions. We just looked at decomposing the resultant force delta F into three stress components, sigma z, tau zx and tau zy on the small area delta A. If we're talking about stress at a point in a material, we have to remember that that material exists in three dimensions, so it is not really a small area delta A but a small volume delta V. If we consider this cubic volume in this Cartesian coordinate space, we can then start to look at the different faces. We have the zx plane with a normal stress sigma y and two additional shear stress components, tau yz and tau yx. We also have the zy plane with normal stress sigma x and shear stress tau xz and tau xy. Of course, we have the reaction stresses on the opposite faces, which for an infinitely small volume will be equal to their opposing face to maintain equilibrium. We get a three by three matrix of normal and shear stress components as shown here to describe the general state of stress in three dimensions. I also want you to remember that although we're drawing this as a cube, it is infinitesimally small. It represents a point. However, in order to visualize the stress components and the direction and the face that they would be acting on, we do draw it as a cube. This stress state in many engineering problems can actually be reduced down to two dimensions. In such cases, our three by three matrix reduces to a two by two matrix with two normal stress components and two shear stress components as shown here. We will often draw this pictorially as shown on the right with a square of width dx and height dy with the stress components acting on it as shown. Now as with the 3D stress state, it's important to remember that this square still represents a volume. In fact, it has a thickness, a differential, an infinitesimal small thickness coming in and out of your screen. However, as we did before, even though it's drawn as a cube or a square, it still represents a point.