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What is the physical interpretation of gradient of a scalar field and directional derivative and its application?
If you think of it as a hill, then the gradient points toward the top of the hill. With the same analogy, directional derivatives would tell the slope of the ground in a direction.
What is Change in elevation divided by difference?
i presume from the question that you are referring to the gradient of a line or road/hill the gradient of a hill is the change in the height (the rise) divided by the hozintal distance (the run) i hope this helps
What is another word for sloping hill?
Incline, slope, gradient, talus...
What is the physical interpretation of gradient of a scalar field and directional derivative?
The elevation of points on a hill is a scalar 'field'. It can have a different value at every point, but each one is a scalar value. Imagine a lumpy bumpy irregular hill, and pick a point to talk about, say, somewhere on the side of the hill. At that point, the directional derivative of the elevation is the rate at which the elevation changes leaving the point in that direction. It has different values in different directions: If you're looking up the hill, then the d.d. is positive in that direction; if you're looking down the hill, the d.d. is negative in that direction. If you're looking along the side of the hill, the d.d. could be zero, because the elevation doesn't change in that particular direction. The directional derivative is a vector. The direction is whatever direction you're talking about, and the magnitude is the rate of change in that direction. The gradient is the vector that's simply the greatest positive directional derivative at that point. Its direction is the direction of the steepest rise, and its magnitude is the rate of rise in that direction. If your hill is, say, a perfect cone, and you're on the side, then the gradient is the vector from you straight toward the top, with magnitude equal to the slope of the side of the cone. Any other vector is a directional derivative, with a smaller slope, and it isn't the gradient.
What is gradient or slope?
Slope is rise/run, or vertical height/horizontal distance. If a hill rises 100 feet high over a horizonatl distance of 1000 feet, it has a slope of 100/1000 = 0.10. Thais one -tenth slope, or one-tenth gradient, which is a 10% grade.
